®l|p  i.  in.  Ml  IGibrarg 

Nnrtl)  (HaroUna  #talF 
lirtitjpraitg 

folio 


li4 
cop. 


This  book  was  presented  by 

-Irs.   Elizabeth  von  Voigtlander 

in  memory  of 

Frederick  von  Voigtlander 


THIS  BOOK  IS  DUE  ON  THE  DATE 
INDICATED  BELOW  AND  IS  SUB- 
JECT TO  AN  OVERDUE  FINE  AS 
POSTED  AT  THE  CIRCULATION 
DESK. 


THE  DESIGN  OF  DIAGRAMS 

FOR 
ENGINEERING  FORMULAS 


[lUlulllj|iiiiijjinHijiiiiii|iiiii||inii||iiinT]nTmT 


HkQrawOJill Book  (h.  Im 

PUBLISHERS     OF     EOOK.S     F  O  R_/ 

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American  Machinist  ^  Ingenieria  Intemacional 
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Industrial  Engineer 


THE  DESIGN  OF  DIAGRAMS 
FOR  ENGINEERING  FORMULAS 

AND 
THE  THEORY  OF  NOMOGRAPHY 


BY 

LAURENCE  I.  HE  WES,  B.  Sc,  Ph.  D, 

MemhcT  of  the  American  Society  of  Civil  Engineers 

Deputy  Chief  Engineer 

V.  S.  Bureau  of  Public  Roads 


AND 

HERBERT  L.  SEWARD,  Ph.  B.,  M.  E. 

Member  of  the  American  Society  of  Mechanical  Engineers 

A  ssociate  Professor  of  Mechanical  Engineering 

Sheffield  Scientific  School 

Yale  University 


FiKST  Edition 


McGRAW-HILL  BOOK  COMPANY,  Inc. 
NEW  YORK:  370  SEVENTH  AVENUE 

LONDON:  6  &  8  BOUVERIE  ST.,  E.  C.  4 

1923 


Copyright,  1923,  by  the 
McGbaw-hill  Book  Company,  Inc. 


PRINTED    IN    THE    UNITED    STATES    OF    AMERICA 


Ya^9f   ^Or^ /^^'^L.^c^.^^^^^--- 


PREFACE 


It  is  intended  in  this  volume  to  present  in  a  prac- 
tical way  the  principles  of  the  design  of  diagrams  or 
nomograms  for  the  solution  of  engineering  and  other 
formulas.  The  usefulness  of  a  diagrammatic  solution 
of  a  formula  is  becoming  increasingly  recognized  and 
it  is  generally  in  proportion  to  the  resistance  of  the 
formula  to  calculation  and  to  the  frequency  of  the 
application  of  the  result  sought.  The  aim  of  the 
present  writing  has  been,  therefore,  not  merely  to  give 
elementary  methods  of  drawing  simple  diagrams  but 
also  to  develop  the  grasp  of  the  reader  so  that  he  will 
be  able  to  analyze  the  more  complex  formulas  of  engi- 
neering practice. 

The  entire  subject  would  only  be  handicapped  by 
attempting  to  avoid  the  use  of  the  third  order  deter- 
minants and  consequently  that  notation  is  introduced 
in  the  third  chapter  and  continued  throughout  the 
book.  A  sufficient  treatment  of  determinants  is 
given  in  Appendix  A  and  is  indispensable  to  those 
who   are  not  familiar  with   that  branch   of   college 


The  use  of  the  projective  transformation  is  men- 
tioned, but  the  reader  may  proceed  independently  of 
that  notion  In  Appendix  B,  however,  is  given  a 
simple  treatment  of  that  subject  sufficient  to  enable 
anyone  who  is  interested  to  understand  its  applica- 
tion to  the  present  theory. 

By  the  determinant  notation  the  identification  of 
given  formulas  with  known  types  is  much  helped 
although  it  is  not  completely  furnished  in  all  cases. 
It  is  hoped,  however,  that  the  necessary  identification 
for  these  cases  has  been  made  much  more  complete 
by  the  introduction  of  an  entire  new  class  of  diagrams 
or  nomograms  which  it  is  proposed  to  call  "Diagrams 
of  Adjustment."  These  diagrams  are  new  and  are 
treated  in  the  last  chapter.  All  other  diagrams 
of  alignment  are  special  cases  of  these  more  general 
types  for  they  may  naturally  be  regarded  as  diagrams 
of  adjustment  in  which  the  adjustment  reduces  to 
zero. 

The  list  of  fifty-four  illustrative  examples  is 
selected  to  avoid  trivial  instances.  It  is  hoped 
that  the  careful  presentation  of  the  general  theory 
of  the  introduction  of  scale  factors  and  units  of  length 
into  the  diagram  will  enable  the  reader  to  produce 


designs  that  are  practical.  For  this  reason  several 
difficult  examples  have  been  worked  out  in  consid- 
erable detail. 

The  geometric  theory  governing  the  position  of 
component  elements  such  as  curves,  lines  or  points 
which  constitute  the  permanent  diagram  must  always 
be  modified  by  the  application  of  certain  limits  of 
accuracy  and  by  a  choice  of  the  range  of  values  of 
the  variables  for  which  the  formula  is  to  be  used. 
The  construction  of  a  permanent  diagram  does  not 
consist  in  the  plotting  of  an  indefinite  number  of 
results  computed  directly  from  the  formula,  but  rather 
in  a  neat  segregation  of  the  several  functions  in  the 
formula  so  that  when  certain  corresponding  scales  are 
plotted  and  suitable  simple  geometrical  constructions 
applied,  a  useful  diagram  results.  The  labor  thus 
involved  is  usually  slight  compared  to  the  resulting 
economy  in  the  use  of  the  formula  for  direct  computa- 
tion. Diagrammatic  representation  of  a  formula  per- 
mits the  immediate  determination  of  the  value  of 
any  variable  and  usually  also  permits  the  determina- 
tion of  the  rate  of  variation  of  any  variable  with 
respect  to  another  variable  when  such  variations  are 
not  readily  determined  or  observed  by  direct  inspec- 
tion of  the  formula. 

The  teaching  of  this  subject  of  diagrammatic  repre- 
sentation of  formulas,  or  Nomography,  at  the  Sheffield 
Scientific  School  for  the  past  nineteen  years  has 
furnished  opportunity  to  the  authors  to  test  its  value 
as  a  supplementary  course  in  applied  mathematics 
and  refined  drafting,  as  well  as  in  practice,  and  con- 
sequently all  unnecessary  theory  has  been  sacrificed. 

A  comprehensive  set  of  problems  is  given  at  the 
close  of  each  chapter  and  many  of  them  may  easily 
be  varied  by  the  choice  of  method  or  of  scale  factors. 

Acknowledgement  is  assuredly  due  to  Professor 
M.  d'Ocagne  whose  fundamental  Traite  de  Nomog- 
raphic doubtless  awakened  the  present  great  interest 
in  this  fascinating  subject  and  whose  own  sympathetic 
interest  in  an  English  exposition  was  expressed 
promptly. 

Laurence  I.  Hewes. 
Herbert  L.  Seward. 
Washington,  D.  C, 
New  Ha%-en,  Conn., 
May,  1923 


CONTENTS 


Page 

Preface v 

Chapter  I. — Function  Scales 1 

1.  The  function  scale 1 

2.  Derivation  of  new  scales 1 

3.  Equations  in  two  variables 5 

4.  Choice  of  scale  factor 7 

Problems 8 

Chapter  II. — Elementary  Diagrams.    .    .  9 

5.  Simple  or  elementary  diagrams.    ...  9 

6.  Scale  factors 9 

7.  Simple  straight  line  diagrams 13 

8.  Anamorphosis 13 

9.  Special  form  of  equation 21 

10.  Hexagonal  diagrams 28 

Problems 30 

Chapter    III. — Alignment    Diagrams    or 

CoLLiNEAR  Nomograms 35 

11.  General  equation  tj^je  and  method  of 

treatment 35 

12.  Diagrams  with  three  parallel  straight 

scales 36 

13.  Diagrams  with  straight  scales  and  two 

only  parallel 43 

14.  Diagrams    of   alignment   with    curved 

scales 49 

15.  Diagrams  of  alignment  with  one  fixed 

point 57 

Problems 62 


Chapter   IV. — Alignment   Diagrams   for 
Formulas    in    More    than   Three 
Variables 65 

16.  Binary  function  scales  and  curve  nets.     65 

17.  Collinear  diagrams  with   two  parallel 

scales  and  one  curve  net 66 

18.  Collinear   diagrams   with   three   curve 

nets 74 

Problems 75 

Chapter  V. — Alignment  Diagrams  with 
Two  or  More  Indices 76 

19.  Diagram  of  double  alignment 76 

20.  Diagrams  with  parallel  or  perpendicular 
indices 80 

21.  Diagrams  for  the  equation  /i  +/2  + 

/3+/4+   .    .    .   +A  =  0 83 

Problems 87 

Chapter  VI. — Alignment  Diagrams  with 
Adjustment 88 

22.  Equations  in  three  variables 88 

23.  Special  forms  of  equations 89 

24.  General  form  of  the  equation  in  three 

variables 93 

25.  Equations  in  more  than  three  variables    94 
Problems 99 

Appenddc  a. — Determinants  of  the  Third 
Order 103 

Appenddc  B. — The  Projective  Transfor- 
mation   105 

Index 109 


FIGURES 

Figure  Subject 

1.  Figures  for  function  scale  yjz 

2.  Figure  for  function  scale  log  z 

3.  Figure  for  the  ordinary  scale 

4.  Figure    for    change    from    scale    factor 

Ml  to  M2 

5.  Figure    for    change    from    scale    factor 

Ml  to  M2 

6.  Figure  for  function  scale  for  a/f{z) 

7.  Figure  for  the  general  projective  scale .  .  . 

8.  Figure  for  the  general  projective  scale .  .  . 

9.  Figure  for  F{z)  =  g^  _^  ^ 

10.  Figure  illustrating  the  measuring  scale, 

log  tan  s 

11.  Figure  illustrating /(z)  =  z  +  sin  2 

12.  Figure  illustrating  scales  converting  inches 

into  hundredths  of  feet 

13.  Figure  illustrating  scales  converting  gal- 

lons into  cubic  feet  of  water 

14.  Figure  illustrating  relation  between  pres- 

sure and  volume  of  dry  steam 

15.  Figure    illustrating    ordinary    Cartesian 

graph 

16.  Diagram  for  the  proportion  of  strength 

in  a  riveted  plate 

17.  Diagram  for  Francis'  Formula  for  stream 

flow  over  a  weir 

17a.  Same  enlarged  near  origin 

18.  Diagram  for  the  solution  of  the  general 

quadratic  equation 

19.  Figure  to  illustrate  the  process  of  ana- 

morphosis   

20.  Diagram  for  the  "External"  of  a  High- 

way Curve 

21.  Diagram  for  the  mean  pressure  of  expand- 

ing steam 

22.  A  second  form  of  diagram  for  the  strength 

of  a  riveted  plate 

23.  Diagram  for  plotting  curves  in  thermo- 

dynamics by  Brauer's  method 

24.  Diagram    for    the    diameter    of    a    shaft 

transmitting  given  horsepower 

25.  Diagram   for  the   approximate   areas  of 

the  segment  of  a  circle 


AND  DIAGRAMS 

Page   Figure  Subject  Page 

1     26.  Diagram    for    the    special    equation 

1  /l+/2+/3  =  0 21 

1  27.  Supplementary  figure  to  Fig.  26 21 

28.  A  second  form  of  diagram  for  Francis' 

2  weir  formula 22 

29.  Diagram  for  friction  head  in  pipe 23 

2     30.  Diagram   for  discharge   of  water  under 

2  above  condition 24 

3  31.  Diagram  for  velocity   of  steam  from  a 

3  turbine  nozzle 25 

.     32.  Diagram  for  wind  resistance  of  an  auto- 
mobile       26 

33.  Diagram  for  inductive  voltage  in  a  parallel 

5  circuit 27 

6  34.  Figure  for  demonstrating  the  hexagonal 

principle 28 

7  35.  Figure  for  demonstrating  the  hexagonal 

principle 28 

7     36.  Hexagonal  diagram   for   the   formula  of 

Fig.  29 29 

7  37.  Hexagonal  figure   for  the  formula  /i  -f 

/2+/3+    .    .    .    +/'.  =  0 30 

8  37a.  Diagram  for  d  = 

10  \ j^ •■    "^^ 

1 1  37b.  Diagram  for  5  =  -7^ — 32 

11  ^  +  3,000  i' 

37c.   Diagram    for    horsepower    in    air    com- 

12  pression 33 

37d.  Diagram  for  Richardson's  equation  for 

13  thermionic  current  from  heated  metals 

in  vacua 34 

15  38.  Figure    for    illustrating    the    diagram    of 

alignment 35 

16  39.  Diagram   for   the   volume   of   excavation 

by  the  method  of  end  areas 37 

17  40.  Diagram  for  the  volume  of  a  torus 40 

41.  Diagram  for  the  exact  area  of  a  circular 

18  segment 41 

42.  Diagram   for  Taylor's   formula   for   tool 

19  pressure 42 

43.  Diagram   for   temperature   correction   of 

20  barometer  readings 44 


FIGURES  AND  DIAGRAMS 


50 


51 


52 


44.  Diagram  for  the  flow  of  steam  through  a 

nozzle,  modified  Napier's  rule 45 

45.  Diagram  for  the  English  automobile  power 

formula 46 

46.  A  third  diagram  for  Francis'  weir  formula     48 

47.  Diagram  for  natural  draft  in  a  chimney 

of  height  //  feet 

48.  Diagram  for  solving  the  quadratic  equa- 

tion z"^  -\-pz  +q  =  0 

49.  Diagram    for    solving    the    cubic    s'  + 

pz  +  q  =  Q _ 

50.  Diagram    for    reducing    stadia    measure- 

ments      54 

51.  Diagram    for    the    hydraulic    radius    of 

trapezoidal  sections 56 

52.  A    fourth    diagram    for    Francis'     weir 

formula 58 

53.  Diagram    for    the    equation    of    thermo- 

dynamics PiFi"  =  P2F2" 59 

53a.  Same  enlarged 60 

54.  Diagram  of  four  straight  scales  for  the 

combined  formulas  of  Fig.  30 61 

55.  Diagram  for  the  flow  through  rectangular 

orifice 62 

56.  Diagram  for  mean  temperature  difference    63 

57.  Figure    for    demonstrating    the    "binary 

scale" 66 

58.  Figure  for  curved  net  and  parallel  scales .  .      68 

59.  Diagram  for  the  complete  cubic  equation     68 

60.  Diagram    for    Flynn's    modification    of 

Kutter'.s  formula 70 

61.  Diagram  for  Bazin's  formula  for  the  flow 

of  water 71 

62.  Diagram  for  the  bond  valuation  formula     73 

63.  Figure  for  three  curve  nets 74 

63a.  Figure  for  the  diagram  of  double  align- 
ment      76 


64.  A  second  form  of  diagram  for  the  hy- 

draulic radius  of  trapezoidal  sections .  .      77 

65.  Diagram    for    Unwin's    formula    for    the 

flow  of  steam 79 

66.  Figure  illustrating  the  theory  of  parallel 

indices 80 

67.  Diagram   for  Lame's   formula   for   thick 

cylinders  (parallel  alignment) 81 

68.  Diagram   for  Lame's   formula   for   thick 

cylinders  (perpendicular  alignment) ...     82 

69.  Figure  for  the  formula /i  +/,  =/3  +/,.     84 

70.  Figure  for  a  modified  form  of  Fig.  69. .  .  .     84 

71.  Figure    for    the    formula  f\-\-  f2+  fz  + 

.    .    .   +/"  =  0 84 

72.  Diagram  for  the  formula  for  the  time  of 

turning  a  piece  of  work  in  a  lathe 85 

73.  Diagram    for    Lewis'    formula    for    the 

strength  of  gear  teeth 86 

74.  Figure  for  three  curve  nets  with  curve 

sets  repeated 88 

75.  Diagram  for  the  quadratic  equation  with 

adjustment  of  index 90 

76.  Diagram    for    the    equation    ZiZi  —  Z3  + 

Vl  +  22^  \1  +  Zi^  =  0 91 

77.  Diagram  for  the  quadratic  equation  with 

use  of  binary  scale 92 

78.  Diagram  for  the  curved  binary  scale 93 

79.  Diagram    for    the    complete    cubic    with 

straight  scales 95 

80.  Diagram  for  the  formula  for  the  length 

of  an  open  belt 96 

81.  Diagram  for  Greene's  heat  flow  formula . .      98 

82.  Diagram  for  the  complete  equation  of  the 

fourth  degree 100 

83.  Figure  for  projective  transformation 104 


LIST  OF  GENERAL  TYPE  EQUATIONS  TREATED 


Type  Page       No. 

a+m 1 

_^  2     21. 

m 


Type 
Xi  = 


bniX 


{H2  -  /ii)x  +  Ml 


bf{z)  +  C 


22. 


^^        ':/(s)  +  <f 

F{z)=^mz)\ 5     23. 

/(z)  =  z  +  sin  z 5 

^="(32)  = /(2i) 5      24. 

/i23  =  0 9,88 

Z1/3  +  Z2g3  +  A3  =  0 13     25. 

/(zl)/3  +  Z2g3  +  ^3=0 14 

/(zl)/3+/(Z2)g3  +  //3=0 14 

/2-/3/i  =  0 16     26. 

/i+/2+/3  =  0 21,36 

/1+/2  +  /34-    .    .    .   +/„  =  0 30,43,84 

.A    gi    1    I 

U     go     1     1=0 35,88 

/3     ^3      1      I  27. 

/ig2  +  /2g3  +  hgx  -  hgy  -  hgo.  -  fig,  =  0 36 

1-1        /i       1     [  28. 

1        {'      ^       =0 36 

I      0     -t       1     I  29. 

si°  =  A'22V 38 

ZiZ-tZi   =  constant 39      ■^• 

/l+/2+/3+/4  =  0 43 

i     0     g:     1 

1     1     gn     1       =  0 47     30. 

h    0     1 

/3(gi-g2)-gi  =  0 47 

//  -  f„'h,'  =  0 47     31. 

0  -/2'         1 

32. 
0 47 


1 

1_ 

\  +  h 

aiZi  +  b 
a^Zi  + 


/■  =  ±z-W 


_    a2Zi  +  &2 

*'  ~  a^Zi  +  63 
1     gi     1 
0      g2      1 

h       g3        1 
g2  +  higl  - 


)    -    g3    =   0. 


dtijfs 


M-2/3  -   Ml(^3   -    1) 

MlM2g3 
M2/3  —  /il(/"3   —    1) 


(M2  —  Mi)«  +  Ml 
0     g,     1 

/2       0  1  =    0.  .  . 

/3       g3       1 
glfs+Ms   -  figl    =   0. 
-1        gl        1 
1         g2        1 
/3       g3        1 
fel  +  g2)    -  /3(gl   -   gs)    -   2g3    =0. 
X   =   b  (^.1  jLj^2)/3jt^(Ml_-lM2) 
(mi  -  M2)/3  +  (mi  +  M2) 

_  2MlM2g3 


(mi  -  M2)/3  +  (mi  +  M2) 

.  g(Mi  +  IX'i)x  +  (mi  -  M2) 

(/ii  -  ^2)*  +  (mi  +  M2) 

2juiiU2y 

(mi  -  M2)-V  +  (mi  +  M2) 

y3g2  =  o 

h       g2       1 

=  0. 


h  1  1 

0     0  1 

</)i(.vy)  =  Si 

</>2(.Vy)    =  Z2 


/l 
/2 
/3. 
1 

0 

/3. 

g2    +/34(gl 
X    =    5 

a;  =  0 


g2)    -  g3  4 


Migi 

M2g2. 


53,55 


^M2/3 


M2/34    ■ 

-1  gl 

1  g2 

/34  g3 


M.(/. 
1 
1 
1 


/34(g. 

-5 


?2)    -    (gl  +  g2)    =  0. 


=  5 


_       (mi   +  M2)/34  +    (mi    —   M2) 
"''  (mi    -    M2)/34  +    (M1  +  M2)' 

yi  =  Migi 

y,   = 


2MlM2g34 
M2)/34  +    (mi  + 


55 


55 


LIST  OF  GENERAL  TYPE  EQUATIONS  TREATED 


No. 

37. 

Type 

/l2      gl2       1 
/34       g3  4        1 
/56       g56       1 

= 

=  0.. 

• 

... 

Pace 
74 

38.  Ao  =  A.     

76,83 

39. 

^  =  ?: 

78 

f        6" 

40. 

1          h      0 
0      -a"     1 
p'        0       1 

=  0 

1 

0 

h     0 

-<?•  1 

0     1 

=  0...  78 

«■; 

'2   -   g,   _   g4  -   gZ 

80 

g2-   gl 

h 

A  -  h 

83 

"^^"■h-jr    g.-g.- 

42. 

42a. 

1        h         0 
/l       gl       1 
h       g2       1 

1     A     0 

/i    gi     1 

/2       g2       1 

=  0 
=  0 

1 

/3 
/4 

1 

g 
g 

h 

g3 
g4 

0 

1 

-/3       1 

-A    1 

0. 

80 

0 83 

43.  yj+y„  -  Jz+Ja 

83 

^           M1P2                     M3M4 
Ml  +  M2           M3  +  M 

83 

4 

45. 

/l2       gl2       1 
/23       g23       1 
/3l       g31       1 

)= 

=  0.. 

88 

48. 


49. 


52. 


/n 

gl2       1 

/23 

g23       1 

h 

g3         1 

h 

gl         1 

h 

g2         1 

u 

g23       1 

In 

gl2       1 

h 

g3          1 

W 

g3'        1 

2  — 

23  +    V 

/n 

gl2       1 

lu 

gl3       1 

h 

gl         1 

Ui 

gii       1 

hi 

gH        1 

i^n 

gmn     1 

/l2 

gl2       1 

/23 

g23       1 

/34 

g34       1 

/l= 

gl2       1 

/23 

g23        1 

/4. 

g45       1 

vrT" 


93,99 


94 


LIST  OF  EXAMPLES  SOLVED 


Example 

1.  lOOsi  =  12z2 

2.  S2  =  7.481zi 

3.  log  Zi   =   Yl  log  Z2. 

4.  _ppri.o65  =  483.... 


5.  S 


30.  z'  +  ^z  +  9  =  0 

Zl.  H  =  R  -  R  sin^  a  +  c  cos  a. 

32.   F  =  K-  sin  2a  +  sin  a 


I.  b  =  T  tan 


P 

6.  5  =  3.33B£r'^ H 

7.  2''  +  /'z  +  ?  =  0 13 

{ - 

9.P„=P,1±^ 14 

10.  D  =  P(l-5) 16 

11.  (tan  «+!)"  =  (tan  |3  +  D 17 

.3/  h.p. 

(r.p.m. 


16.  V  =  223.8 V(l-  Y){H,-H,). 

17.  P  =  0.003a52 

d        . 

0.78r 


18.      E  =  0.232  log 
,F' 


19.  H  =  o.a^^j  .,„ 

20.  A- F  =  (/>!  +  /(s) 36 

21.  F  =  2.4674M= 39 

H       V2RH  -  F- 


33.  P 


A(6  +  k  cot  <|)) 


Pace 

.   51 
.   53 

.  53 
.  55 


6  +  2AV1  +  cot2  ,j> 

34.  q  =  3.335E''^ 57 

35.  PF»  =  C 57 

36.  z'  +  aiz^  +  ajz  +  as  =  0 67 


4i.e 


37.  F 


1.81132      0.00281 

+        n       ^5 


1  +  [41.6603  + 

87 


0.00281 -[    w 
5      WR 


Vrs. 


Vrs- 


0.552  + 


'1  - 


40.  cos  lz  =  .h  ^         , 

^         \       cos  L  cos  h 


41.  R  =  H 


/cos  5  cos  (5 
cos  Z,  CO 

1  +  A  cot  <A 


1  +  2A:  cosec 
42.  W  =  87.5 


Z)/>4^ 


Z.   1  + 


3.61 


43.  F 

44.  D  =  (f 


Vrs.... 

S  -  P 

'S  +  P 


23. 


CF''D^'' 


24.       //  =  /!i[l  -  0.000101(/i  -  /)] 

po.97  j^^ 

W^ 

£)W5 

12    

27.      Q  =  3.335Zr'= 47 

1 


25.  F  = 

26.  HP 


Parallel  indices 

Perpendicular  indices. 


43 


49.  Z1Z2  -  Z3  +  vT+Z2=- V  1  +  2i'  =  0 

50.  2*  +  fliz  +  02  =  0 

51.  z'  +  aiz*  +  floz  +  as  =  0 

52.  L  =  RU  +  29)  +  r( 


28.      D  =  (.52)  (14.7)  W 


49 


-461  4-  60       461  +  n   ■  ■  • 
29.  Rules  for  writing  determinants 51 


53.  T  = 

54.  z'  +  ais^  +  a: 


«(r,  -  r2)-|- 


-  29)  4-2Ccose 94 

97 

+  (J4  =  0 99 


DESIGN  OF  DIAGRAMS  FOR 
ENGINEERING  FORMULAS 


CHAPTER 


FUNCTION  SCALES 


1.  The  Function  Scale. — In  the  construction  of 
permanent  diagrams  for  the  numerical  solution  of 
formulas  or  equations  it  is  constantly  necessary  to 
use  a  scale  on  which  lengths  are  proportional  to  the 
values  of  a  function  of  a  single  variable.  The  value  of 
the  variable  is  the  important  item  so  it  is  written 
beside  the  point  determined  by  the  corresponding 
value  of  the  function.  Thus,  let  it  be  required  to 
construct  the  scale  for  the  function  y/z  for  values 
of  z  ranging  from  0  to  5,  then  if  unity  is  represented 
by  2  inches,  the  length  of  the  scale  will  be 

L  =  2X\/5=2X  2.236  =  4.472  inches 

and  its  end  will  be  marked  5.  The  number  4  will  be 
written  at  the  end  of  the  segment  04  =  2\/4  =  4 
inches,  3  at  the  end  of  the  segment  03  =  2V3  = 
3.464  inches  and  so  on.     See  Fig.  1. 


VT 


a  scale  of  the  function /(s) .  If  it  should  happen  that 
for  necessary  values  of  s  the  lengths  OM  are  incon- 
veniently large  or  small,  these  lengths  may  be  modified 
by  the  introduction  of  a  scale  factor  /x  and  laid  off  as 

OM  =  m/(2) 

Thus  in  the  example  worked  out  above  where  the 
linear  unit  is  one  inch,  ;u  =  2.  The  linear  unit  adopted 
on  a  drawing  may  also  be  called  a  modulus.  Diagrams 
for  engineering  formulas  involve  more  than  one  function 
scale  in  a  figure  and  in  such  a  diagram  the  modulus 
{i.e.  unit  of  length)  is  usually  adopted  and  suitable 
scale  factors  selected,  as  explained  in  Article  4.  (See 
also  Article  6,  Chapter  II.) 

If /(s)  reduces  to  s  itself,  the  resulting  scale  (Fig.  3) 
is  the  ordinary  scale  of  the  draftsman.  This  scale 
will  be  called  the  ordinary  scale. 


I  I       I      I    I    1   I  I  I  I  I  I  I  II  llllll        I        I        I       I I I  I  I  II  ll  I  I  I  I  I 


2 


hiiA 


Fig. 


A  very  familiar  example  of  a  function  scale  is  found 
in  the  common  slide  rule  where  the  function  is  log  s, 
and  the  lengths  are  laid  off  proportional  to  the 
logarithms  of  the  numbers  z,  as  in  Fig.  2. 


I     I     I    I 


I    I    I   I    I    I  I  M  lllllllllllllillllllillllll      I      I     I     I    I    I    I    I 


2.  Derivation  of  New  Scales. — As  there  is  often 
considerable  computation  necessary  in  the  construc- 
tion of  a  function  scale  it  is  desirable  to  make  use  of 
several   graphical   methods  which   help   to  establish 
the  scales  of  new  functions  from 
scales  already  made. 

(a)  To  establish  the  scale  of 


Fig.  2. 

The  notation /(:),  and  more  often  simply/,  will  be 
used  to  denote  any  function  of  a  single  variable. 
Starting  from  an  initial  point  0,  if  the  successive 
lengths  OM  =  f{z)  are  laid  off  and  the  points  M 
inscribed  with  the  successive  values  of  s,  there  results 


a+Iiz) 

from  that  of/(s),  where  a  is  any  constant,  it  is  merely 
necessary  to  move  the  inscribed  values  of  z  forward 
or  backward  the  distance  a  according  as  a  is  positive 
or  negative. 

(6)  The  next  simple  case  is  the  change  of  the  scale 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


factor  of  the  function  scale.  Suppose  the  scale  of 
/(c)  is  established  on  the  line  MN  (sometimes  called 
the  support  of  the  scale),  Fig.  4,  with  the  scale  factor 


M 


H 


M' 


N' 


/A' 


Ml.  The  scale  factor  may  be  changed  to  /i2  by  simply 
drawing  a  hne  M'N'  parallel  to  MN  and  projecting 
the  division  points  of  the  scale  on  MN  to  the  line 
M'N'  from  a  point  0  such  that 

OH   _Mi 
OH'      M2 

Naturally  this  construction  will  also  serve  to  set 
up  with  the  same  modulus,  the  scale  of  af{z)  from 
that  of /(z).  A  common  case  occurs  when  the  scale 
log  2  is  given  and  it  is  desired  to  obtain  the  scale  of 
log  2"  or  a  log  z.  In  hydraulic  formulas  this  oppor- 
tunity is  often  presented. 

(&')  Other  methods  of  handling  the  same  problem 
are  available  especially  where  the  supports  for  the  two 
scales  are  not  parallel.  Suppose  as  before  that  the 
scale  iox  f{z)  has  been  set  up  on  the  hne  MN,  and  the 
scale  for  af{z)  is  desired  on  a  Hne  MN'  with  the  same 
modulus.     In   Fig.    5    the   two   supports  are  shown 


Q       N 


meeting  at  an  angle  0.  If  QQ'  is  drawn  at  any  con- 
venient location  such  that 

MQ       1 

MQ'  ~  a 

it  is  simply  necessary  to  project  the  points  from  the 
line  MN  to  the  Hne  MN'  by  Hues  paraUel  to  QQ' . 
In  certain  cases  it  may  be  desirable  to  project  the 


points  on  MA''  parallel  to  PP'  which  is  perpendicular 
to  MN.     In  this  case  (assuming  a  >  1) 

1 

cos  0  =  ~ 
a 

Cases  (a)  and  {b)  combined  furnish  a  method  of 
constructing  a  scale  for  the  function  a  +  bf{z). 
The  order  of  carrying  out  the  work  is  immaterial. 

(f)  To  set  up  the  scale  of  ttt  from  the  scale  of /(z) 

given  on  the  line  MN,  Fig.  6,  the  procedure  may  be 
as  follows: 

Draw  a  circle  with  the  center  at  M  and  with  radius 
Va,  and  let  MP  =  f{z).  Now  if  PT  is  tangent  to 
the  circle,  and  TP'  is  drawn  perpendicular  to  MN, 
then  MP'  MP  =  a.  Hence  when  the  points  P' 
have  been  marked  with  the  same  values  of  z  as  are 
found  at  the  corresponding  points  P,  the  new  scale  is 
complete.  If  P  is  within  the  circle,  thenP'  is  without 
and  is  found  by  drawing  TP  perpendicular  to  MN  and 
then  drawing  the  tangent  TP'  to  locate  P' . 


By  making  use  of  (a),  {b)  and  (c)  combined,  the 
scale  of 

a 

bfiz)  +  c 

can  also  be  obtained  from  that  of/(z). 

{d)  The  functions  in  the  preceding  three  cases  are 
special  cases  of  the  more  general  case, 

af{z)  +  b 
cfiz)  +  d 

where  ad  —  be  is  not  equal  to  zero.  To  establish  the 
scale  -for  this  function  of  z  from  that  of /(z)  it  is  suffi- 
cient to  project  the  division  points  of  the  scale  of /(z) 
from  a  point  P,  called  a  ray  center,  to  a  line  M'N' 
making  an  angle  ((>  with  the  support  MN,  Fig.  7. 

The  most  practical  method  of  establishing  the  scale 
for  F{z)  from  that  of /(z)  in  this  case  is  to  compute  the 
location  of  two  points,  Zi  and  Z2,  on  the  scale  for  F{z) 


Fiz) 


FUNCTION  SCALES 


by  substituting  in  the  given  function  F{z)  two  conven- 
ient values  of  s  and  plotting  the  resulting  values  of 
F{z) .  Then  the  scale  for/(s)  can  be  placed  at  an  angle 
<i>  to  the  scale  of  F{z)  and  corresponding  values  of 
^1  and  Zi  on  /(s)  and  F{z)  can  be  joined  by  rays  the 
intersection  of  which  determines  the  ray  center  P. 
The  angle  4>  should  be  so  chosen  that  the  intersections 
of  all  rays  with  both  scales  are,  as  far  as  possible,  not 
too  oblique,  and  also  so  that  the  ray  center  P  will  not 


be  located  at  too  great  a  distance  from  the  scales. 
Sometimes  the  ray  center  P  may  be  located  between 
the  two  scales.  It  is  always  well  to  check  the  position 
of  P  by  a  third  ray  through  another  pair  of  correspond- 
ing points  on  the  two  scales.  Due  to  the  above  par- 
ticular properties  the  scale  for  F{z)  =  —j-  /  -.         ,  is 

known  as  a  projective  scale. 

{d^)  It  is  possible  to  compute  the  coordinates,  m 
and  n,  of  the  ray  center  P  from  the  values  of  a,  b,  c, 
and  d  in  F(z) .  The  graphical  method  of  the  preceding 
paragraph  is  more  direct  but  occasionally  it  may  be 
desirable  to  check  the  location  of  the  ray  center  P. 
If  the  supports  of  the  two  scales  are  used  as  coordinate 
axes  (Fig.  8),  the  expressions  for  the  oblique 
coordinates  of  P  may  be  found.  These  will  be 
ad  -  be  cf(zo)  +  d 

c{cf  (zo)  +d)  c 


Fic.  8. 

It  is  assumed  that  the  scales  intersect  at  corresponding 
points  designated  as  so  on  /(z)  and  so  on  F{z) . 

af(z)  +  h  ^,  ^  dFiz)  -  b 


Consider  the  projections  obtained  when  a  ray  from  Zi 
on/(s)  parallel  to  the  scale  of  F{z)  is  drawn  and  another 
ray  from  ^2  on  F{z)  parallel  to  the  scale  of /(s)  is  drawn. 
In  Fig.  8,  Si  must  make  F{z^  =   oo . 

.■./(=,)  =  -^ 


Again 

Now 
But 
Hence 
Also 


F{z, 


22  must  make/(s2) 
I  =  F(z2)  -  F(z,)  = 

^  g/(2o)  +  b 
^       cfizo)  +  d 

_  ad  —  be 
'  -  eiefizo)  +  d) 

,  =  f{z,)  -  f{zo)  = 


(a  -  eF(zo)) 


cf(zo)  +  d 


The  coordinates  m  and  n  are  thus  determined  by 
the  value  z  =  zo  and  this  leaves  the  choice  of  the 
angle  <j>  so  that  P  can  always  be  placed  in  the  acute 
angle  between  the  supports  of  the  scales. 

For  example  let 

If  it  is  desired  to  plot  the  scale  for  this  function  by  trial 
rays  as  described  in  section  (d)  above  compute  values 
of  F{z)  for  given  values  of  z: 


z 

F{z) 

Points 

0 

7 

A 

Yi 

3.6 

B 

1 

2.75 

C 

3 

1.9 

D 

_2 

0.2 

E 

Using  a  sheet  of  coordinate  paper  (Fig.  9)  the  scale 
for  F{z)  is  partially  plotted  along  the  horizontal  axis 
from  the  origin  0  in  points  A,  B,  C,  D  and  E.  If 
at  any  point,  say  A,  a  scale  for/(z)  which  in  this  case 
is  z  itself  (or  the  ordinary  scale)  is  constructed,  the  rays 
to  B,  C,  D  and  E  through  correspondingly  numbered 
points  on  the  two  scales  all  intersect  at  P,  the  ray 
center.  Then  the  remaining  points  on  F{z)  could  be 
graphically  determined  as  fully  as  desired  by  pro- 
jecting points  from  f{z)  to  the  support  for  F{z). 
Here  4>  has  been  taken  as  90°. 

To  illustrate  the  use  of  the  method  in  section  (rf')  it 
is  noted  that 


b  =  7 


Since  F{z)  = 


cm  +  d 


/(=) 


-cF{z)  +  a 


In  choosing  a  value  for  zo  we  are  simply  selecting  a 
common  point  on  the  two  scales  /(s)  and  F{z).    If 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


—  q. 

:4 

E 

^ 

^ 

_  3 

:3 

"_  o 

1 

/ 

/- 

^ 

r" 

^('»^ 

/ 

9 

/. 

/ 

-    1 

/ 

/; 

-^ 

1 

0 

-2 

F( 

z) 

3/ 

i^A 

c:^ 

r^ 

■ 

A 

=  0 

"         " 

-£:r 

^=^ 

H 

f^ 

re- 

B 

-     m 

— ____ 

- 

r 

^^ 



= 

p-l 

\ 

\ 

\ 

^^ 

^ 

-  .? 

^1  +  7 

izfl 

\ 
\ 

"    _7 

\ 

"  -4. 

~   -5 

1 

1 

\ 

\ 

--& 

FUNCTION  SCALES 


2o  is  chosen  as  zero,  it  means  that  at  the  point  on 
^■(2)  which  is  inscribed  zero  there  is  constructed  a 
scale  for  /(c)  having  its  zero  in  coincidence  with  the 
zero  of  F{z).  Substituting  in  the  expressions  for  m 
and  n,  noting  that  /(co)  =  20  =  0  there  are  obtained 
the  values 

m  =  -5.667 

n  =  -0.333 
Using  the  point  of  coincidence  of  the  two  scales  as  an 
origin  (point  A  of  Fig.  9)  the  ray  center  F  may  be 
located  by  measuring  the  values  of  m  and  n,  as  indi- 
cated in  the  figure.  The  scale  for  ^(2)  can  then  be 
determined  as  fully  as  desired  by  projection  from 
f{z)  as  before. 

If  /(20)  had  been  chosen  as  unity  in  the  preceding 
paragraph,  the  values  of  m  and  n  would  be 

m  =  -1.417 

n  =  -1.333 
and  the  scale  for 7(2)  would  have  been  erected  at  point 
C  with  its  unity  at  C.     The  ray  center  would  then  be 


(/)  The  graph  of  a  function  may  be  used  to  advan- 
tage in  setting  up  the  scale  of  that  function  whenever 
the  graph  may  be  drawn  mechanically  either  wholly 
or  in  part.     Thus  for  example  if 


/(=) 


2  +  sin  2 


the  graph  OP  A  may  be  drawn,  Fig.  11.  If  OM  repre- 
sents 2,  ON  is  of  length  2  -f  sin  2  and  if  N  is  marked 
2,  there  is  secured  the  desired  function  scale  on  OY. 
In  the  figure  the  curve  OP  A  was  drawn  by  adding  the 
ordinates  of  the  two  curves  B  and  C;  B  represents 
J'{z)  =  2  and  C  represents  f"(z)  =  sin  2.  Since  the 
curve  C  may  be  obtained  by  the  construction  indicated 
in  dotted  lines  on  the  right,  the  entire  work  of  con- 
structing the  scale  of  J{z)  =  2  -f-  sin  z  can  be  done 
graphically.  The  graphical  method  becomes  espe- 
cially important  when  the  analytical  expression  for  a 
function  is  not  known.  This  is  usually  the  case  when 
the  graph  of  a  function  is  obtained  from  experimental 
observations. 


0.3  0.4 

1 .1  1 1  II  l.l  1,11.1 1 


log  z 
0.5       0.6 


0.7    0.8  0.9  1.0 


2,0 


25  30 

log  tan  z 

Fic.  10. 


located  at  P'  as  shown  and  a  new  set  of  rays  would 
determine  the  same  points  on  ^(2)  as  before, 
(f)  The  scale  of 

Fiz)=f[^iz)] 
may  be  obtained  by  using  the  scale  of  7(2)  as  a  measur- 
ing scale  as  follows.  The  quantity  (t>{z)  plays  the 
same  part  in  the  new  scale  as  did  2  in  the  original  scale 
of  f{z).  Given  2,  the  value  of  <j>{z)  can  be  computed, 
and  regarding  it  as  2  the  corresponding  length  can  be 
picked  out  on  the  scale  of  f(z) .  This  length  is  the 
corresponding  length  on  the  new  scale  and  is  inscribed 
with  the  original  value  of  2.  Thus,  for  example,  from 
the  scale  of  log  2  there  can  be  determined  at  once  the 
scale  of  log  tan  2.  There  are  in  the  trigonometric 
tables  values  of  tan  2  for  the  values  of  2  desired;  then 
the  points  on  the  logarithmic  scale  are  selected  which 
are  marked  with  these  values  of  tan  2.  These  lengths 
are  then  laid  off  on  a  new  line  as  support  and  their 
end  points  are  marked  with  the  values  of  2  (not  tan  2). 
(See  Fig.  10  and  Example  11.) 


3.  Equations  in  Two  Variables. — Consider  now  a 
formula  involving  two  variables, 

=2=/(20 

If  on  one  side  of  a  line  there  is  constructed  the  scale 
for  the  function  /(zi)  and  on  the  other  side  the  ordin- 
ary scale  for  22  with  the  same  modulus  and  starting 
with  corresponding  values  at  the  same  point,  then 
any  pair  of  values  which  satisfy  the  above  equation 
are  found  opposite  each  other  on  the  two  scales.  Thus 
is  realized  by  a  diagram  a  numerical  solution  of  the 
equation.  To  illustrate,  if /(2)  =  5,  2  is  found  on  the 
function  scale  opposite  5  on  the  ordinary  scale,  and 
so  on. 

An  obvious  modification  of  this  principle  will  per- 
mit the  construction  of  a  diagram  yielding  all  solutions 
of  an  equation  or  formula  of  the  form 

F(s2)=/(20  (1) 

Construct  the  scale  for  ^(22)  on  one  side  of  a  line  as 
support  and  the  scale  for  f{zi)  on  the  other  side  and 


DESIGX  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


read  the  value  of  22  opposite  the  value  of  21  correspond- 
ing. The  scales  must,  of  course,  start  at  correspond- 
ing values  a  and  b  such  that  F{a)  =  f{h) ,  and  have  the 
same  modulus.  This  case  is  useful  when  the  formula 
is  awkward  to  solve  for  either  variable.  The  follow- 
ing simple  examples  will  serve  as  illustrations : 


Example  2. — The  number  of  gallons  z-i  in  Z\  cubic 
feet  may  be  written 

22  =  7.48121 

or  log  22  =  log  7.481  +  log  2, 

After  laying  out  the  logarithmic  scale  for  22  on  the 


ii.n 

. 

348°  I 

/I 

33&°- 
324°- 

/ 

/      / 

317°- 

/ 

/ 

300- 
288°- 

1G4- 
252"- 
140°^ 

/ 

^ 

/ 
/         y 

/ 

84- 

/ 
/ 

TZ"- 

IL 

r 

/ 
/ 

/ 

<o\j  — 
48- 

/ 

Sfc- 

/ 

24°- 

L 

/ 

1  1  1  1  1 

sC-- 

— .---  - 

,-_^ 

1  1  1  1  1 

Mill 

\ 

r ^s 

\'l°- 

M 

1  1 

1  1 

1 1 1 

1    1    1    1    1    1    iNU-r  1 Mill 

1    1    1       A 

0 

30° 

b 

0° 

'90°' 

120" 

150°      180°       210°      240° 

210° 

300" 

330°      360° 

formula    for   converting   inches 
a   foot  and  conversely  may  be 


Example  1. — The 
into  hundredths  of 
written 

IOO22  =   122, 
where  21  denotes  hundredths  of  a  foot  and  22  inches. 
Clearly  all  that  is  needed  is  to  have  a  Une  divided  on 
one  side  for  inches  and  on  the  other  side  for  hundredths 
of  a  foot  as  in  Fig.  12. 


upper  side  of  the  line  in  Fig.  13,  construct  the  logarith- 
mic scale  for  21  with  its  point  marked  1  opposite  7.481 
on  the  upper  scale.  Both  scales  have  the  same  modu- 
lus and  may  be  transferred  from  a  slide  rule  or  a 
logarithmic  rule. 

Example  3. — log  21  =  ^^  log  22  is  the  equation 
solved  on  the  slide  rule  when  square  roots  of  22  are 
found. 


FUNCTION  SCALES 


Example   4. — The  empirical  formula 

p^.  1.065  _  4g3 

giving  a  relation  between  the  pressure  and  volume  of 
one  pound  of  dry  saturated  steam,  may  be  written, 

logP=  log  483-  1.065  log  F. 

Lay  out  a  logarithmic  scale  of  any  convenient  modulus 
on  the  lower  side  of  the  line  as  in  Fig.  14,  for  the  scale 
of  P.  At  the  point  483  on  this  scale  is  found  the  unity 
of  the  V  scale,  since  when  V  =  1,  P  =  483.  The  scale 
factor  of  the  logarithmic  scale  for  V  will  be  1.065  and 
the  scale  wiU  increase  in  the  direction  opposite  to  that 
of  P.  This  scale  may  be  constructed  by  the  method 
of  Article  2(ft)  or  2(6')- 


Hundredfhs 


\\•,<,\\<,<.'^^/l>lff^\\^\^\\^'K^^^^^^^^^ 


1        8 
Inches 
Fig.  12. 


An  equation  of  the  form 


where  K,  a  and  b  are  constants,  may  be  represented 
readily  by  the  use  of  logarithmic  scales.  The  expo- 
nents a  and  b  really  become  the  scale  factors  and  one 
scale  is  translated  a  distance  log  K. 

Another  method  of  treating  an  equation  in  two 
variables 

22  =  /(2i) 

is  to  make  use  of  the  ordinary  cartesian  graph.  In 
Fig.  15  let  C  be  the  graph  of  the  above  equation 
referred  to  the  axes  OX  and  OY.  For  a  given  Zi  say 
OM,  draw  MP  perpendicular  to  OX  and  from  P  drop 
a  perpendicular  PN  to  OY.  Then  ON  is  the  desired 
value  of  22.  Coordinate  or  cross-section  paper  would 
ordinarily  be  used  for  this  type  of  diagram. 

Diagrams  representing  equations  in  two  variables 
are  used  more  to  supplement  the  usefulness  of  more 
complicated  diagrams  than  to  afford  in  themselves 
a  means  of  solving  equations  in  two  variables.  In 
later  examples  it  will  be  found  that  many  of  the  scales 
are  graduated  for  two  quantities,  such  as  cubic  feet 
per  second  and  gallons  per  minute,  on  the  same  line. 
While  only  one  of  these  quantities  may  appear  in  the 
formula  for  which  the  diagram  is  drawn,  the  addition 
of  the  other  often  increases  the  usefulness  of  the 
diagram. 

4.  Choice  of  Scale  Factor.— The  construction  of 
the  scale  of  a  function  with  a  suitable  scale  factor  /x 
is  an  essential  operation  in  the  design  of  any  perma- 
nent diagram  for  numerical  solutions.  The  length 
L  of  the  desired  scale  is  limited  by  the  size  of  the  paper 
and  must  satisfy  the  equation 

L  =  4Ab)  -  f{a)] 


—  § 


8 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


where  a  and  b  are  the  limiting  values  of  the  variable 
and  n  is  the  scale  factor  to  be  selected.  There  is 
usually  some  choice  of  these  limiting  values  a  and  b, 
and  as  more  than  one  function  scale  is  involved  the 
relation  of  the  various  scales  in  the  diagram  must  be 
carefully  studied  in  advance.  The  use  of  the  scale 
must  be  kept  in  view  and  the  graduations  arranged 
so  that  interpolations  by  eye  will,  when  possible, 
yield  one  figure  beyond  the  required  accuracy.  When 
some  portion  of  a  non-uniform  scale  is  to  be  most 
frequently  used  that  portion  should  be  given  the 
advantage  of  the  larger  graduations  by  the  methods 

y 


developed  in  Chapter  III,  as  for  example  in  the  stadia 
formula 

It  is  always  desirable  to  check  various  points  on  a 
new  scale  by  double  calculations  and  by  various  known 
characteristics  of  the  function  such  as  the  magnitude 
and  uniformity  of  the  rate  of  increase  within  a  given 
interval  of  the  variable.  The  accuracy  of  the  finished 
diagram  should  also  be  checked  by  characteristics 
of  the  given  formula  and  by  various  numerical 
examples. 

Problem  1. — Construct  a  diagram  showing  the  relation 
between  kilowatts  and  horsepower. 


Problem  2. — Construct  a  diagram  showing  the  relation 
between  circular  pitch  and  diametral  pitch  of  gear  teeth. 

Problem    3. — Construct    the    projective    scale    of    the 
function 

1.7  log  s  +  6.5 
2.4  -  0.84  log  z 
from  a  logarithmic  scale. 
Problem  4. — Construct  a  scale  for  values  of  <p  from  0° 

to  135°  for  the  function  tan 


Fiz) 


[j  +  *]- 


Problem  5. — Plot  the  function  scale  for  ^-  between   the 

limits  2  =  2  and  2  =  50  upon  a  hne  12  inches  long. 
Problem  6. — Plot  a  scale  for  the  function 
1  +  sin  z 


1  —  sin  z 
for  values  of  z  from  0°  to  30°. 

Problem  7. — Plot  a  scale  for  the  function 
the  line  OX. 

Problem  8. — Establish 
3/+  1 


_ffH_ 

H^'  4-  2 


function  scales       _        and 


on  the  X  and  Y  axes  respectively  and  show  that 


corresponding  values  of  /  determine  values  of  the  coordi- 
nates that  locate  points  on  a  straight  line.  What  is  the 
equation  of   this  hne   in   cartesian  coordinates? 

Problem  9.— Plot  the  scale  for  0.38  F'  '^  starting  from 
a  definite  point  0  on  the  line  OX. 

Problem  10. — Construct  a  scale  for  the  law 

between  the  limits  h  =  I  and  h  =  100. 
Problem  11.— Construct  a  scale  for  ( 0.405  -|-  -^^ — ) 

between  the  limits  h  =  0.2  and  h  =  1.4. 

Problem  12. — Construct  a  diagram  for  the  velocity  v 
due  to  an  adiabatic  heat  drop  A/7  for  steam  from  the 
expression  v  =  223.8v'a^. 

Problem  13.— Construct  a  diagram  forP  '"^F  =  327.7. 


CHAPTER  II 


ELEMENTARY  DIAGRAMS 


5.  Simple  or  Elementary  Diagrams. — An  equation 
or  formula  involving  three  variables  Zi,  Zi,  and  S3  may 
be  denoted  by 

/(2lZ2Z3)    =    0 

or  more  briefly  by 


h 


(2) 


One  of  the  main  objects  of  this  volume  is  to  develop 
the  construction  of  a  permanent  diagram  for  solving 
Equation  (2) .  Such  a  diagram  should  determine  any 
third  variable  when  two  are  given  and  it  is  frequently 
called  a  nomogram.  A  natural  method  would  be  to 
let  3i  and  Z2  represent  independent  variable  coordinates 
X  and  y  and  plot  the  family  of  curves 

fixyz,)  =  0 

In  such  a  diagram   the  parameter  23  which  varies 
from  curve  to  curve  should  be  the  variable  whose 
values  from  the  nature  of  the  given  problem  increase 
by  fixed  intervals. 
Example    5. — To   illustrate,   consider   the   formula 

^  -      p 

which  gives  the  proportion  of  strength  5  remaining 
in  a  plate  at  a  riveted  joint,  where  P  is  the  pitch  and 
D  the  diameter  of  the  rivet  holes,  both  in  inches.  In 
this  formula  P  and  5  have  almost  any  values  (within 
certain  limits)  while  D  usually  varies  by  sixteenths 
of  an  inch.     Take  then 

P  =  X    and     S  =  y 

leaving  D  as  the  parameter  of  the  system  which  will 
require  the  least  number  of  curves.     There  results 
then  the  simple  curve  equation 
X  -  D 


with  values  of  D  ranging  from  }^  to  l}4  inches  by 
eighths  or  by  sixteenths  if  desired.  Substituting 
D  =  J^  in  the  equation  of  the  curve  family  gives 

x-y2 


Plot  this  curve  and  mark  it  D  =  J^.  One  could  then 
proceed  to  plot  each  of  the  curves  for  values  of  D 
equal  to  ^,  %,  1}4,  etc.,  and  thus  obtain  the  system 
of  curves  for  D  as  shown  in  Fig.  16.  Since,  however, 
for  any  two  successive  values  D'  and  D"  of  D  the 
corresponding  ordinates  yi  and  y^  for  a  given  abscissa 
Xi  are 

xi-  D'        ,            xi-  D" 
yy  =  — ~ and   y^  = :.- 


it  follows  that 


D'      xi 


D"  -  D' 


that  is  to  say  for  the  same  abscissa,  the  increments  of 
any  ordinate  for  equal  increments  of  D  are  equal. 
It  is  necessary  then  to  plot  only  the  extreme  curves 
for  D  =  3^  and  D  =  1)4,  and  divide  the  portion  of 
each  vertical  line  which  is  included  between  them, 
into  the  same  number  of  equal  parts.  In  Fig.  16 
the  ordinates  were  divided  into  eight  equal  parts. 

To  find  5  from  the  diagram  when  P  and  D  are 
given:  Find  the  intersection  of  the  ordinate  at  the 
point  on  the  X  axis  corresponding  to  the  given  value 
of  P  with  the  curve  marked  with  the  given  value  of 
D  and  then  read  the  required  value  of  5  horizontally 
on  the  Y  axis.  For  example,  if  P  =  "iM  inches  and 
D  =  %  inch,  find  the  ordinate  3J^  and  the  curve 
D  =  1i  intersecting  at  the  point  A ,  which  is  opposite 
the  value  75  on  the  Y  axis.  The  figure  may  be 
entered  with  any  two  variables  and  the  third  found. 
Suppose,  for  example,  it  is  required  to  determine  what 
pitch  will  be  required  with  ^^  inch  rivets  to  give  an 
efficiency  of  80  per  cent.  From  the  point  80  on  the 
Y  axis  run  horizontally  to  the  right  as  far  as  the  curve 
D  =  %,  then  run  vertically  to  the  .Y  axis  and  the 
required  value  of  P  is  3.75  inches. 

6.  Scale  Factors. — The  scale  factors  used  when  the 
ordinary  scales  for  21  and  Zi  are  constructed  on  the 
axes  of  coordinates  need  not  be  the  same.  It  is  usual 
to  have 

x  =  niZ\     and    y  =  M222 


10  DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 

100 


90 


80 


10 


s:     GO 


t^       50 


c£^       40 


20 




^::r 

^ 

^ 

^ 

^^ 

^ 

^ 

5 

^ 

§ 

p/ 

/ 

K 

7 

^ 

^ 

'^ 

^ 

^ 

^ 

/ 

/ 

^\ 

^ 

^ 

q' 

/ 

// 

// 

// 

>; 

/ 

/ 

/ 

// 

^ 

// 

/// 

'// 

'// 

1 

' 

RIVETED  MINTS 
Sfrength  of  plate  a  f Joint 
compared  withsohd plate 

/ 

?  3  4  5 

P  =    Greatest  Pikh  of  Rivets  in  inches 


ELEMENTARY  DIAGRAMS 


11 


Then   for   the   equation  /123 
becomes 


0    the   curve   system 


This    equation    is    the   one   from   which    the   family 
of  curves  S3  is  plotted  to  ordinary  and  equal  scales 


where  H  is  the  head  of  the  water  on  the  crest  of  the 
weir  in  feet,  B  the  width  of  the  weir  in  feet,  and  q 
the  discharge  in  cubic  feet  per  second.  Of  these  three 
variables  the  breadth  B  is  the  one  which  is  most  likely 
to  be  expressed  in  even  numbers,  so  it  is  chosen  for  the 
parameter  of  the  system  of  necessary  curves.     The 


11 
II 

10  90 

9 

8 

80 
7 

6 

S    10 

4 

3 

60 
2 

1 

A 

/ 

/ 

/ 

/      / 

/ 

/ 

/ 

/ 

/ 

/       / 

/  / 

/ 

V 

1/ 

/ 

/ 

/       - 

/ 

^  / 

/ 

// 

/ 

/ 

/ 

/ 

/ 

/ 

/    . 

'  / 

/ 

/  . 

'    / 

/ 

/ 

f — 

7- 

/ 

// 

/    / 

y 

y 

-J^o 

// 

/ 

y 

y 

//\^  y 

y 

y 

/ 

/ 

/ 

/ 

'// 

\^ 

X  ^ 

y 

/^ 

'  y  y 

/- 

^ 

/ 

/ 

/ 

/ 

//y 

y  y 

'>' 

^ 

^ 

/ 

yy 

/  - 

^-'^ 

//y 

// 

^  J 

^  \ 

:b'^ 

^-^^ 

/ 

/ 

/ 

/ 

/ 

y^ 

yy^ 

^ 

,-^ 

^.^  1 

1-- 

,--^ 

^ 

/ 

y 

y>^^ 

-^ 



. 

f/ 

y 

/ 

/ 

y 

/ 

^-> 

^ 

-^ 

■^;^;;^ 

" 

^ 

- — -^ 

)               0.1 

0.2          0.3          0.4-          0.. 

CO) 

U  bO 

l^ 

(/ 

y 

7^ — 

7^ 

/ 

y- 

y 

Diagrams  for 
FRANCIS  'WEm  FOR 

f 

{/ 

/ 

y 

/ 

/ 

y 

MULA 

30 

/ 

V 

/ 

/ 

y 

V 

y 

y^ 

^ 

/ 

y. 

/ 

/ 

y 

y 

^ 

^ 

/ 

y. 

yy 

y 

/ 

y 

^ 

A, 

/^ 

^ 

20 

/y 

yy 

/ 

y 

y 

^ 

^^ 

^ 

/y 

^ 

^ 

^ 

^ 

^ 

^ 

1 

^ 

10 

y 

^ 

/^ 

^ 

^ 

^ 

^ 

^ 

>^ 

>^ 

^ 

^ 

^ 

^ 

^ 

I 

— 

n 

^^ 

^ 

^ 

fe 

^=^ 

"^ 



0.1      Q3      0.4     0.5      0.6      0.7 


0.8      0.9       1.0      I.I 

Head  H  in  feet 

Fiu.  17. 


1.3       1.4       1.5 


.8      1.9      2.0 


on  both  axes.  It  is  necessary  to  introduce  such  scale 
factors  when  one  of  the  independent  variables,  say 
2i,  varies  through  a  greater  range  than  the  other  z^. 

Example  6. — In  Fig.  17  there  is  shown  a  diagram  for 
Francis'  formula  for  the  discharge  of  water  over  a  weir 
without  end  contractions, 

q  =  Z.ZWH^"- 


head  H  is  usually  less  than  two  feet  while  with  B  = 
10  the  discharge  q  runs  up  to  94.2  if  H  =  2.  So  the 
values  of  q  run  through  a  range  of  numbers  about  50 
times  as  great  as  the  corresponding  values  of  H. 
Accordingly  it  will  be  desirable  to  plot  the  scale  for  H 
with  a  scale  factor  which  is  about  50  times  that  used 
for  the  scale  of  q. 


12 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


Diagram  for  the  quadratic  z^  +  pz  +  q  =  0 
Fio.  18. 


ELEMENTARY  DIAGRAMS 


13 


Let 


then 


Mii?     and 
-  =  3.3351 


©" 


is  the  equation  of  the  system  of  curves  for  B.  In 
Fig.  17  Ml  was  taken  as  5  and  m  as  0.1  so  that 

y  =  0.2978a;'^5 
is  the  equation  of  the  curve  system  referred  to  natural 
scales  on  the  coordinate  axes.  Since  each  value  of 
0.2978.T**  is  multiplied  by  B  it  is  necessary  to  plot 
only  the  curve  for  5  =  10  and  divide  each  of  its 
ordinates  into  10  equal  parts  to  obtain  the  entire 
system  of  curves  for  B.  In  Fig.  17a  the  curves  near 
the  origin  are  shown  drawn  to  a  larger  scale. 

7.  Simple  Straight  Line  Diagrams. — The  labor  of 
constructing  diagrams  such  as  were  given  above  is 
considerable  unless  the  family  of  curves  is  easily 
plotted.  The  curves  can  be  made  straight  lines  when- 
ever Equation  (2)  has  the  form 

z,f{z,)+z,giz,)  +  h{z,)  =  0 
or  more  briefly 

S1/3  +  S2g3  +   /^3   =    0  (3) 

where /a,  gz  and  h^  are  any  functions  of  S3,  may  or  may 
not  be  alike  and  frequently  reduce  to  constants  includ- 
ing zero.  The  use  of  the  same  letter  to  denote  func- 
tions of  different  variables  in  what  follows  will  not 
necessarily  mean  that  the  functions  are  the  same 
although  such  may  sometimes  be  the  case.  In  general, 
for  example,  /(si)  or  /i  will  not  denote  the  same 
functions  as/(z3),  etc. 

Whenever  Equation  (2)  has  a  form  which  may  be 
reduced  to  the  above  form  (3)  by  suitable  transforma- 
tions, set 

X  =  ix\Z\     and     y  =  mZi 
and  Equation  (3)  becomes 

Xti-ifs  +  3'Mlg3  +  MlM2/?3  =   0 

which  determines  a  family  of  straight  lines  marked 
with  corresponding  values  of  S3.  Equations  in  three 
variables  such  as  Equation  (2)  occur  very  frequently  in 
engineering  practice  and  are  of  particular  interest  here. 
Example  7. — The  general  quadratic  equation 
z'-  +  pz^q  =  Q 
if  .r  =  p  and  y  =  q  becomes 

s=  -f  xs  -I-  y  =  0 
This  is  a  straight  line  system  and  the  original  Equation 
is  of  the  type  (3)  where 


When  as  many  lines  of  the  system  have  been  drawn 
as  the  diagram  will  comfortably  admit  at  suitable 
intervals  of  z  it  is  seen  that  for  moderate  values  of  the 
coefficients  p  and  q  it  is  possible  to  solve  any  quad- 
ratic by  reading  the  roots  written  on  the  lines  passing 
through  the  corresponding  intersection  point  of  the 
lines  X  =  p  and  y  =  q.  It  will  be  necessary  to 
interpolate  for  all  the  quantities  p,  q  and  z.  See 
Fig.  18. 

8.  Anamorphosis. — It  is  possible  in  a  large  class  of 
equations  which  do  not  fall  under  the  type  of  Equation 
(3)  to  reduce  the  needed  family  of  curves  to  straight 
lines.  It  will  first  be  shown  how  this  may  be  done 
graphically  with  a  single  curve  and  then  the  method 
will  be  extended  to  apply  to  a  family  of  curves. 


and 


=  1, 


Suppose  there  is  given  in  Fig.  19  a  single  curve  C 
corresponding  to  some  particular  value  of  the  quantity 
S3  in  the  equation 

/(S,S2S3)   =   0 

This  curve  may  be  changed  to  a  straight  line  L 
which  will  serve  equally  well  to  determine  either  of  the 
corresponding  quantities  Si  and  22  as  foUows.  Draw 
any  oblique  line  AB  and  let  every  point  P  of  the 
curve  C  be  projected  horizontally  into  a  corresponding 
point  Q  upon  the  line  L.  Now  inscribe  N,  the  foot 
of  the  ordinate  of  Q,  with  the  value  of  21  which  is 
found  at  M  on  the  X  axis.  After  a  sufficient  number 
of  points  have  been  treated  in  this  way  the  curve  C 
may  be  erased,  also  the  old  scale  of  Si  and  then  the 
diagram  serves  to  determine  the  corresponding  values 
of  Si  and  so  for  the  value  of  S3  originally  used.  This 
process  was  called  by  Lalanne  "Anamorphosis."' 
What  has  been  done  changes  the  scale  on  OX  from 
the  ordinary  scale  for  Si  to  a  certain  function  scale. 
To  see  this  it  is  only  necessary  to  notice  that  the 
length  ON  is  always  a  function  of  the  length  OM. 

A  logical  extension  of  the  above  principle  to  all  the 
curves  S3  of  a  given  family  is  desirable.  For  this 
purpose  it  will  be  necessary  from  the  given  equation 

/(s,s.S3)  =  0 
to  select  a  function  x  of  Si  such  that  when  y  =  ^222 
the  entire  family  of  curves  corresponding  to  values 
'  L.  Lalanne,  Annales  des  Ponis  el  Chaussies,  1846. 


14 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERIXG  FORMULAS 


of  23  shall  be  straight  lines.     That  is,  it  is  necessary 
to  change  the  original  equation 

/(212223)  =  0 
by  virtue  of  the  relations, 

X  =  Mi/(si)  y  =  H1.Z2 

into  a  linear  equation  in  x  and  y.     A  necessary  and 
suflScient  condition  is  that  the  original  equation 

f{z,z.z,)  =  0 
may  be  reduced  to  the  form 

/(Sl)/3  +  S2g3  +h   =    0  (4) 

For  in  this  Equation  (4)  if  Zi  and  zj  are  eliminated  there 
results 

Xpifi  +  Vyuigs  +  MiM2/'3  =    0 
which  is  the  equation  of  a  family  of  straight  lines  to 
be  inscribed  with  values  of  23- 

Equation  (4)  is  of  the  form  that  will  yield  straight 
lines  when  a  function  scale  is  used  on  the  A'  axis  only. 
If.  however,  the  ordinates  also  are  made  to  depend  not 
simply  on  22  but  on  a  function  of  22,  as  7(22),  there 
results  a  method  of  treating  equations  of  greater  gener- 
ality.    Set  therefore 

X  =  /ii/i     and     y  =  ti-ifi 
then  when  the  Equation  (2),/i23  =  0,  has  the  form 

/(Sl)/3+/(22)g3+//3    =    0  (5) 

it  will  yield  a  system  of  straight  lines  for  the  values 
of  23,  by  virtue  of  these  relations. 

This  is  the  principle  underlying  the  use  of  "logarith- 
mic cross-section  paper"  for  plotting  an  equation  in 
two  variables.  This  paper  is  a  cross-section  paper 
ruled  with  logarithmic  scales  on  the  axes  instead  of 
with  the  ordinary  scales.  Any  equation  in  two  vari- 
ables which  has  the  form 

21-S2'  =  A-, 
for  example,  where  a,  h  and  K  are  constants,  may 
immediately  be  given  the  form 

a  log  zi  +  &  log  Z2  —  log  A'  =  0 
by  taking  the  logarithm  of  both  sides.     The  resulting 
equation  has  the  form  (5).     When  therefore 

X  =  log  zi         y  =  log  Z2 
the  above  equation  reduces  immediately  to 

ax  -{-  by  —  log  K  =  0 
which  is  a  straight  line  equation  for  the  ordinary  cross- 
section  paper.  Or  in  other  words,  if  corresponding 
values  of  Z\  and  Z2  determined  from  the  original  equa- 
tion are  plotted  directly  on  the  logarithmic  cross-sec- 
tion paper,  the  resulting  coordinates  are  proportional 
to  the  corresponding  logarithms  and  the  graph  is  a 
straight  line. 


The  exponents  a  and  h  determine  the  slope  of  the 
resulting  straight  Une;  i.e.  —  v- 

This  principle  when  used  inversely  is  of  great  value 
in  determining  the  unknown  exponents  for  an  empiri- 
cal formula  when  a  sufficient  number  of  points  are 
plotted  on  the  logarithmic  cross-section  paper  from 
actual  observation  and  are  found  to  determine  closely 
a  straight  line. 

Equation  (5)  is  a  very  general  type  equation  and 
includes  a  large  number  of  formulas  of  engineering. 
Such  formulas  will  frequently  require  algebraic  and 
sometimes  logarithmic  transformations  in  their  form 
before  they  can  be  identified  with  the  type  by  inspec- 
tion. It  will  be  seen  that  the  corresponding  diagrams 
consist  essentially  of  three  systems  of  straight  lines 
and  that  two  of  these  systems  are  parallel  to  the  axes, 
determined  by  function  scales  on  the  axes. 

The  foregoing  Equation  (5)  is  not  the  most  general 
equation  in  three  variables  whose  diagram  can  be 
constructed  by  three  straight  line  systems  provided 
no  restriction  is  placed  on  the  nature  of  the  systems. 
Such  an  equation  is  best  expressed  in  determinant  form 
but  can,  however,  be  treated  by  much  more  elegant 
methods  than  those  of  the  present  chapter. 

Example  8. — The  "external"  or  distance  from  the 
intersection  of  two  tangents  to  the  curved  line,  in  high- 
way or  railroad  surveying  is  given  by  the  formula 

Z)  =  r  tan  7 
4 

where  T  is  the  length  of  the  tangent  and  /  the  acute 
angle  of  intersection.  In  the  field  it  is  often  desired, 
before  finally  determining  either  T  ox  h  for  a  given 
angle,  to  try  several  pairs  of  values,  and  the  diagram 
given  in  Fig.  20  is  convenient. 

The  formula  is  in  the  form  of  Equation  (4)  where 


tan^=/(2,),  6 


22,    T    =Ug3=    -  1,    //3 


so  that  if 


M26 


there  results  the  radial  line  system 

V  X  „ 


limit  of  b  is  taken  as  18  feet  the  diagram  of  Fig.  20 
can  be  drawn  with  in  =  0.3  and  M2  =  60. 

Example  Q. — The  mean  pressure  Pm  of  steam 
expanded  from  an  initial  pressure  Pi  according  to 
the  law  PV  =  constant,  is  given  by  the  formula 


ELEMENTARY  DIAGRAMS 
q  nvfcaaxxa 


15 


q  iVNHaxxa 


16 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


if  measured  above  a  back  pressure  of  absolute  zero. 
R  is  the  ratio  of  expansion. 

1  +  log.  R 
"  R 


is  taken  as/i  and  Pm  as/2  then/3  =  P\,  gz 


Mean  Pressure  of  fxpanded  Sfeam 
according  fo  the  law  pv^p,  V/ 

Pm  -  Absolute  Mean  Pressure 
P,=        ^,  Injfial     '> 

/p-  Ralio  of  Expansion  =}L 


•1  and 


Since  all  the  lines  Pi  pass  through  the  origin  it  is 
necessary  to  locate  but  one  point  on  each  line  to  draw 
the  system.  Such  points  are  very  simply  determined 
by  the  intersections  of  the  radial  lines  with  the  line 
a;  =  1  parallel  to  the  Y  axis.  In  general  when  a 
system  of  radial  Unes  y  =  mx  is  to  be  plotted,  set 


4O0- 


0^^ 


Ratio  of  Expansion 
Fig.  21. 


^3  =  0  showing  that  the  above   equation  is  in  the 
form  of  (5) 

/2-/3/l    =    0 

Accordingly  let 

1  +  log.  R 
x  =  ,. ^ 

and  y  =  ti.iPm. 

so  that  there  results  a  family  of  radial  straight  lines 


(See  Fig.  21.) 


a;  =  1  so  that  y  =  w.  In  the  present  case  the  scale 
determined  on  the  line  a;  =  1  is  an  ordinary  scale 

whose  scale  factor  is  — •     Of  course  beyond  the  limits 

of  the  paper  the  radial  lines  cannot  intersect  the  line 
a;  =  1  and  if  it  is  necessary  to  draw  additional  lines 
they  may  be  determined  by  their  intersections  with  a 
line  parallel  to  the  Y  axis  at  any  convenient  distance. 

P  —  D 

Example  10. — The  formula  5  =  — „ —  of  Example 

5  may  be  written 

Z)  =  P(l  -  5). 


ELEMENTARY  DIAGRAMS 


17 


If  /i  =  (1  -  5),  /o  =  Z?,  /a  =  P,  g3  =  -1,  /ra  =  0  it      it  becomes 
is  in  the  form  of  Equation  (5) 

hh-h  =  o 

Accordingly  let        x  =  ^{1  —  S) 
y  =  i^iD 

y 


ny  —  X  =  Q  Z.S  shown  in  Fig.  23. 


givmg  i^  "  ■^ 

as  shown  in  Fig.  22. 


The  scales  on  the  axes  are  readily  plotted  by  the 
method  of  Article  2  (e);  i.e.,  for  a  given  a  or  /3  look 
up  the  value  of  the  natural  tangent,  add  one,  and  find 
the  resulting  quantity  on  a  logarithmic  scale,  inscribing 
the  point  with  the  value  of  a  or  /3  used. 


Pihh  of  holes  in  inches 


100        % 


% 


85  80  75  70  65 

Efficiency  in  percent 


It  will  be  noticed  that  the  graduations  of  the  5  scale 
on  the  X  axis  increase  toward  the  origin,  since  the 
function  is  (1  —  S). 

Example  11. — The  expression  (tana  +  1)"  =  (tan 
/3  +  1)  is  useful  in  plotting  exponential  curves  of  the 
typePF"  =  constant,  in  thermodynamics  by  Brauer's 
method.     If  written 

n  log  (tan  a  +  1)  -  log  (tan  /3  +  1)  =  0 
with 

X  =  log  (tan  /3  +  1) 

y  =  log  (tan  a  +  1) 

2 


Example  12. — The  formula  for  the  diameter  {d) 
of  a  shaft  to  transmit  a  given  horsepower  (h.p.) 
at  a  given  speed  (r.p.m.)  is  of  the  form 


'i, 


h.p. 


.p.m. 

If  the  allowable  stress  for  a  steel  shaft  is  taken 
13,500,  the  constant  c  has  the  value  2.87;  hence 
1 


d^  =  (2.87) 'h.p. 


r.p. 


If  a  reciprocal  scale  is  used  on  the  A'  axis  and  a  cubic 


18 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


scale  is  used  on  the  Y  axis  a  family  of  straight  lines  Example  13. — The  approximate  formula  for  the  area 

for  values  of  h.p.  results.  A  of  the  segment  whose  height  is  H  of  a  circle  of  radius 

Ris 


Let 
and 


y  =  uid^ 


A  = 


"-    l2R 


4b  - 

7 

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(Tanfi+I) 

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¥'k' 

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10 

-_ 

20  ZS  50  35  40 

Volues  of  (3  J  decrees 

Fig.  23. 


45 


SO  that 


(2.87)'h.p.ci; 


or  9^2  -  32/2^?'  +  9.728Z/^  =  0 

This  equation  is  in  the  form  (5)  with/i  =  A"^  and/2  = 

as  shown  in  Fig.  24.     A  second  set  of  underscored      R  so  that  '\{  x  =  A''-  and  >>  =  /?  a  family  of  straight 

graduations  for  d  and  h.p.  have  been  added,  covering      lines  for  E 

a  larger  range  of  numbers.  %x  -  ZIR^y  +  9.728^^  =  0 


ELEMENTARY  DIAGRAMS 


19 


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20 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


^.     oi    o 
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ssLioui  ui  snipD^  J.0  sanpy^ 


ELEMENTARY  DIAGRAMS 


21 


would  result.     This  equation  is  difficult  to  plot  and 
the  lines  are  poorly  located  for  accurate  reading. 

If  the  ratio  -^  =  A'  is  used  the  equation  becomes 
_  4       V2A-- 0.608 


Then  with  x  =  iufi 
results  a   radial   system 


/ii/1-  and  y  = 


-■"Ji  =  i^2R*  there 
2K  -  0.608 


shown  in  Fig.  25. 

9.  Special   Form   of   Equation. — The    case  where 
Equation  (5)  has  the  simple  form 


/1+/.+/3 


(6) 


is  of  special  importance.  It  gives  rise  to  a  system  of 
parallel  straight  lines,  since  if  a;  =  mi/i  and  y  =  M2/2, 
the  equation  becomes 

IX2X  +  idiy  +  ixiix-ifa  =  0 

This  system  of  lines  may  be  dispensed  with  if  their 
common  normal  through  0  is  drawn  and  on  it  the 
function  scale  for  =3  established.  The  function  scales 
for  Zi  and  S2  must  be  constructed  on  the  X  and  Y 
axes  respectively  as  before.  The  diagram  then 
consists  essentially  of  three  function  scales  whose 
supports  intersect  at  0.  It  is  read  by  finding  the 
unknown  value  of  2  where  a  line  through  the  intersec- 
tion of  the  two  perpendiculars  at  the  given  values  of 
s  on  their  respective  axes  meets  perpendicularly  the 
scale  of  the  unknown  2.     See  Fig.  26. 


Fig.  26. 

Since  the  three  lines  necessarily  perpendicular  to  the 
respective  scales  meet  at  constant  angles  they  maybe 
scratched  on  a  transparent  sheet  which  when  properly 
oriented  on  the  drawing  will  enable  the  unknown 
values  to  be  read  rapidly.  For  ordinary  work,  how- 
ever, a  diagram  having  the  cardinal  values  of  all  three 
straight  line  systems  drawn  in  is  found  to  be  the  best 
arrangement. 

Example  14. — The  formula  for  the  weir  discharge 
used  in  Example  6 


may  be  brought  into  the  form  (6)  by  taking  the 
logarithm  of  both  sides.     There  results 

log  q  —  log  3.33  —  log  B  —  ^2  log  H  =  0 
Here  if  log  q  =  Ji  and  log  H  =  fi'it  is  seen  that 

/3=  -log  3.33  -  log  5 
Set  X  =  111  log  H  and  y  =  /is  log  q.     Then  the  equation 
of  the  parallel  Unes  for  B  is 

^  -  ^  -  -  log  3.33  -  log  5  =  0 

/*2  2  All  * 

These  lines  may  best  be  drawn  if  the  common  normal 
to  the  system  is  first  drawn  and  numbered  with  the 
values  of  B  at  the  points  of  intersection  with  the 
parallel  lines.     To  do  this  it  is  necessary  to  determine 


the  angle  a  of  Fig.  27  (above)  and  the  correspond- 
ing function  scale  on  the  normal.  The  angle  in  the 
present  example  is  126°  52'  12"  and  the  lines  B  inter- 
sect the  normal  at  distances  from  the  origin  determined 
by  the  function 

|M.[log3.33  +  log5] 

The  completed  diagram  is  shown  in  Fig.  28. 

In  general  when  an  equation  is  of  the  form  (6)  and 
the  resulting  system  of  lines  for  23  is  given  by  the 
Equation 

HlX  -\-  jui  V  +  M1M2/3  =  0 
this  last  equation  may  be  put  into  the  normal  form 

X  cos  a  -h  y  sin  a  —  p  =  0 
where 

and  sin  a 


Vmi"  +  M2"  "VViM-mT" 

and  where  the  scale  on  the  normal  is  determined  by 
the  function  fi  with  the  scale  factor 


Vyur  +  112' 

TM.86 

Example   15.— The  formula  H  =  0.38  ^y;^  gives 

the  friction  head  H  in  feet  per  1,000  feet  of  water  flow- 
ing in  a  pipe  of  diameter  d  with  a  velocity  of  V  feet 
per  second.     In  logarithmic  form  the  equation  is 
log  H  +  1.25  log  d  -  log  0.38  -  1.86  log  F  =  0 


22 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


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DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


ELEMENTARY  DIAGRAMS 


25 


li  X  =  ixi  log  H  and  y=  ti^  log  d,  there  results  a  system 
of  parallel  lines  for  V 


+  1.25^ 


log  0.38-  1.86  log  V  =  Q 


Figure  29  shows  the  completed  diagram  with  /U2  = 
1.25  and  mi  =  1-0.  The  normal  bisects  the  angle 
between  the  axes  and  the  scale  on  it  is 

P  =  -^  (log  0.38  +  1.86  log  V) 

If  the  line  system  for  V  is  to  be  drawn,  it  is  of  course 
not  necessary  to  draw  the  normal  since  the  lines  of 
the  system  cross  the  X  axis  in  points  determined  by 
the  scale 

p  =  log  0.38+  1.86  log  V 


upon  eliminating  log  V  between  the  two  logarithmic 
equations  there  results 

1.86  log  Q  =  1.86  log  ^^^11^  +  4.97  log  J  +  log  H 

and  since  x  =  log  H,  y  =  1.25  log  d,  the  system  of 
parallel  straight  lines  for  Q  is 

^  +  iS  ^  =  1-86  log  Q  -  1.86  log  ^^ 

The  necessary  lines  are  added  to  Fig.  29  in  Fig.  30. 
The  angle  for  the  system  Q  is  75°  53'. 

Example  16. — The  velocity  V  with  which  a  jet  of 
steam  issues  from  a  turbine  nozzle  having  a  friction 
factor  Y  is 

V  =  223.8 V(l  -  Y)(H,  -  H~i) 
where  {Hi  —  Ho)  is  the  "Heat  drop"  or  number  of 


V=J(2t  Veloci+L)  in  f+.pcrsec 


wn 


O      G>      O     o   '^     O 


s 


s 


5 


^ 


^ 


S:S 


5 


s 


^=^ 


^ 


^: 


100       no       no      130      140     ISO    160   ITO    180  190  200  210  220  230 140  2S0 260 TO 280  Z90 300    320     M   3M   330    400 
H,-H2=  Heat  Drop, B.+.U. 

Diagram  for  Steam  Jet  Velocities,  V  =  223.8^(1  -Y)  {Hi-  Hi) 
Fig.  31. 


The  discharge  Q  is  equal  to  the  velocity  of  flow 
multiplied  by  the  cross-section  of  the  stream.  For  a 
circular  pipe  of  diameter  d  the  discharge  is 


id'-V 


It  is  possible  to  supplement  the  diagram  of  Fig.  29 
by  new  lines  which  will  give  the  discharge.  The 
example  illustrates  a  general  method  available  for  use 
when  four  variables  occur  in  this  way  in  two  equations. 
Since 

log  Q  =  log  0.7854  +  2  log  J  +  log  V 
and 

log  H  +  1.25  log  d  -  log  0.38  -  1.86  log  V  =  0 


British   thermal   units   of   energy   available.     Figure 
31  shows  a  diagram  for  this  formula  with  the  following 
analysis : 
log    (ffi   -    Hi)   +   log   (1    -    F) 

h  2  log  223.8  -  2  log  F  =  0 
If 

«  =   Ml  log  (^1  -  ^2) 

y  =  fj.2  log  (1  -  F) 
the  parallel  lines  for  V  have  as  their  equation 

HiX  +  Miv  -  MiM-2[2  log  F  -  2  log  223.8]  =  0 
The  normal  is  located  from 


T,  and  sin 


V  Ml"  +  M2" 


26 


DESIGN  OF  DIAGRAMS  FOR  E\GI\EERL\G  FORMULAS 


12     23,    24    25     2&    21    28    29  50  31    52   35  54  35  36  57  38  59  40 
Fig.  32. 


ELEMENTARY  DIAGRAMS 


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DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


and  the  scale  on  it  is 
M1M2 


+  Al2 


^  [2  log  F  -  2  log  223.8] 


The  origin  is  not  shown  on  the  diagram. 

Example  17. — An  empirical  formula  giving  the 
number  of  pounds  of  wind  resistance  R  in  an  automo- 
bile offering  a  square  feet  of  wind  resisting  area  at  5 
miles  per  hour  is 

R  =  0.003a52 

Passing  to  logarithms 

log  a  +  2  log  S  +  log  0.003  -  log  i?  =  0 
If  X  =  til  log  a 

y  =  y-i  log  R 


then 


Ml  M2 

the  scale  on  the  normal  is 


+  2  log  5  +  log  0.003  =  0 


P  =    ,  V    — :,  [2  log  S  +  log  0.003] 

V  Ml"  +  M2" 

The  diagram  is  shown  in  Fig.  32. 

Example  18. — In  Fig.  33  is  shown  a  diagram  includ- 
ing parallel  straight  lines,  for  the  Equation 

£  =  0.232  log  ^^ 

which  gives  the  inductive  voltage  E  per  ampere  per 
mile  of  double  wire  for  alternating  currents,  where  r  is 
the  radius  of  the  wire  and  d  is  the  spacing,  both  in 
inches.  As  the  size  of  the  wire  is  usually  expressed 
by  the  gauge  the  latter  was  used  in  constructing  the 
diagram.  To  correct  E  for  various  frequencies  the 
constant  must  be  varied;  the  present  diagram  is 
drawn  for  both  25  and  60  cycles.  It  is  not  necessary 
to  pass  to  logarithms  in  order  to  bring  this  equation 
into  a  form  similar  to  type  Equation  (6) 

0;|2  =  logJ-log0.78r  =  0 

"  "^  =  '^'0232 

y  =  U2  log  d 


and 


^  +  log  0.78  r  =   0 


the  third  system  is 

/*1  A*2 

10.  Hexagonal  Diagrams. — For  Equation  (6)  above, 
the  resulting  equation  for  the  lines  of  the  variable 
23  may  be  given  a  special  form  by  setting  mi  =  M2 
when  the  range  of  the  values  of  Zi  and  z^  permits. 
The  scale  factor  for  the  Z3  scale  on  the  normal  reduces 

1  1 

then  to  ~~7^.    The  factor  ~7^  may  be  dispensed  with 

by  choosing  the  axes  for  the  Zi  and  22  scales  at  an 
angle  of  120°  and  establishing  the  Zg  scale  on  the 


bisector  of  this  angle.  It  can  be  proved  from  Fig.  34 
that  if  from  any  point  P  perpendiculars  are  drawn  to 
three  scales  there  shown  the  following  geometric 
relation  holds 

OMi  +  OMi  =  OM3 


This  relation  is  easily  seen  by  observing  that  in 
Fig.  35,  AMi  =  M^B  so  that  20M^  =  OA-\-OB,  but 
OA  =  2OM1  and  OB  =  20Mi,  whence  the  relation 
above 


0 
Fig.  35. 

If  now                          OMi  =  m/i 

OM2   =    m/2 

OM3   =    m/3 

it  follows  always  that 

/l+/2=/3 

for  the  values  of  2  found  at  the  corresponding  points 

Ml,  M2,  M3. 

This  form  of  diagram  is  called  the  hexagonal  form 
from  the  fact  that  the  lines  involved  are  the  diagonals 
of  a  hexagon. 

Example  19. — The  formula  of  Example  15 
71.86 


H 


may  be  readily  represented  by  a  hexagonal  diagram  if 
written 

(log  H  -  log  0.38)  +  1.25  log  d=  1.86  log  V 
Figure  36  shows  the  completed  diagram.     While  the 
scale  factors  of  all  three  scales  must  be  the  same,  the 
coefficients   1.00,    1.25,   and   1.86  determine  the  unit 
length  of  the  scales.     The  constant  log  0.38  in  the  H 


ELEMENTARY  DIAGRAMS 


29 


30 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


function  shows  that  the  logarithmic  scale  for  H  must 
be  moved  to  the  left  until  0.38  is  at  the  origin. 

The  hexagonal  diagram  may  be  supplied  with  a 
suflScient  number  of  scales  to  solve  equations  of  the 
form 

/1+/2+/3+   .    .    .   +f„  =  0  (7) 

Write  the  equivalent  system 

//0+/3    =    //I 


/r„-3+/„- 


/n 


On  a  suitably  inscribed  diagram  enter  with  21  and  22 
and  obtain  a  temporary  point  M  on  a  blank  scale. 
Then  from  the  point  where  the  z^  perpendicular  cuts 


fitf^-,f^*f^*fs'0 


the  perpendicular  from  M  drop  a  perpendicular  to 
locate  N  and  proceed  in  this  way  until  s„  is  reached. 
The  arrangement  of  scales  for  n  =  5  is  shown  in  Fig. 
37.  Another  treatment  of  Equation  (7)  will  be  found 
in  Article  21  of  Chapter  V. 

Problem  1. — The  illustrative  examples  of  this  chapter 
(5  to  18  inclusive)  may  in  most  cases  be  represented  by 
diagrams  of  types  other  than  those  used.  Investigate 
all  feasible  tjqjes  for  each  formula  given. 

Problem  2. — The  volume  V  of  the  frustrum  of  a  cone  of 
height  h  is 

V  =  j^h[D'  +  Dd  +  d'] 

where     D    and     d    are     the    diameters    of    the    bases. 
Using  D  and  d  as  si  and  33  show  how  the  system  of  curves 
V 

for  S3  =  -,  may  become  a  family  of  concentric  circles  and 

construct  the  diagram. 


Problem  3. — Boussinesq's  appro.ximate  formula  for  the 
perimeter  of  an  ellipse  L  with  semi-axes  a  and  b  is 

L  =  ^[%(a  +  6)-  y/^b\ 

Show  that  with  a  and  b  as  21  and  =2  the  curves  for  zz  =  L 
may  become  circles  tangent  to  both  coordinate  axes  if  a 
suitable  angle  is  chosen  for  YOX. 

Problem  4. — Draw  all  feasible  diagrams  for 

^  =  e'^ 
Ti 

the  ratio  of  belt  or  rope  tensions  Ti  and  T^  for  a  coefficient 
of  friction  /  and  an  angle  of  wrap  d. 

Problem  5. — Determine  the  corresponding  formula  when 
a  set  of  observations  of  two  variables  result  in  a  parabola 
symmetrical  to  the  Y  axis  when  plotted  on  logarithmic 
cross-section  paper. 

Problem  6. — Construct  a  diagram  for  the  cubic  equation 
z'  +  pz  -\-  q  =  0  similar  to  that  of  Example  7,  page  13, 
with  regular  scales  for  p  and  q  on  the  axes. 

Problem  7. — The  capacity  of  a  silo  is  given  by  K.  J.  T. 
Eckblaw  as 

d'    /h^- 
\20 


C  = 


2M 


256  V20 

where  C  is  the  capacity  in  tons,  h  the  height  in  feet  and  d 
the  diameter  in  feet. 

(a)  Construct  a  diagram  using  parallel  straight  line 
systems. 

(b)  Construct  a  diagram  using  a  radial  straight  line 
system. 

Problem  8. — F.  W.  Taylor  gives  the  expression  for  the 
pressure  upon  a  cutting  tool  when  cutting  cast  iron 

p  =  cd'^''  F^' 
where  P  is  in  pounds,  D  is  the  depth  of  cut  in  inches  and  F 
is  the  feed  in  inches.     The  quantity  C  is  taken  as  45,000 
for  soft  cast  iron  up  to  69,000  for  hard  cast  iron.     Con- 
struct a  convenient  diagram. 

Problem  9.— The  expression  P„  =  3.463Pi(i?-"  -  1)  is 
used  in  determining  the  mean  effective  pressure  P„,  when 
air  is  compressed  from  an  initial  absolute  pressure  Pi 
pounds  per  square  inch  and  R  is  the  ratio  of  the  final  to  the 
initial  pressure.  Devise  a  diagram  with  parallel  straight 
lines. 

Problem  10. — In  problems  involving  compound  interest 
the  expression  i?  =  (1  +  >■)"  is  the  basis  of  all  such  com- 
putations.    Devise  a  useful  diagram  for  this  expression. 

Problem  11. — Devise  and  construct  a  convenient 
diagram  which  may  be  used  to  determine  the  correct 
revolutions  per  minute  for  pieces  of  work  of  various 
diameters  (in  inches)  when  certain  cutting  speeds  (in 
feet  per  minute)  are  desired  in  various  rotary  machines. 

Problem  12. — Look  up  the  formula  by  Grashof  for  the 
flow  of  air  through  orifices  and  construct  a  diagram  for  use 
only  within  the  limits  for  which  the  formula  is  applicable. 

Problem  13. — Construct  a  diagram  for  the  two  formulas 
of  Example  15  using  ordinary  logarithmic  cross-section 
paper  with  equal  scale  factors  on  the  axes.  Plot  H  on 
OX  and  Q  on  OY. 


ELEMENTARY  DIAGRAMS 


31 


d-  DIamc+er  o-f  Sha-f-f  in  inches 

T  6  5 


20 


^0  40  SO    inches 

/L  -  Overhangs  Disiance  befweencenierofbearinc^ 
and  cenfer  of  crank  pin. 


-10 


t-30 


Diagram  for  equaiibn    c(=  iJs.f  F(L  -i-yL^+R ^) 

^^ 
Diameter  of Shaff  for  Combined  Bending  and  Tw/siihg 

Place  poinh  of  dividers  on  values  of  L  and  R.  Wifti  L  as  center 
swing  arc  to  fioHzonta  I  axis j  project  verticallg  to  value  ofFj 
horiiontallu  to  value  off^)  verficallg  to  "d"  ttie  required 
diameter  of  shaft. 

FiQ.  37a. 


32 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


Diagram  giving  safe  unif  stress  ibr 
PLATE  GIRDER  WEBS  WITHOUT STIFFEMERS 
From  Coopers  formula  S= — ^?- 

3000f 


ELEMENTARY  DIAGRAMS 


33 


34 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 

Problem  15. — Construct  a  diagram  for  the  expression 
H  =  0.0274F2  +  0.0141Z,Fi" 


Problem  14. — Construct  a  diagram  for  the  formula 

X  =  2^/(80 +  741. Hog  7)10-"^ 

which  gives  the  inductive  reactance  x  in  ohms  of  a  trans-  for  the  friction  loss  H  in  condenser  tubes  L  feet  long  when 

mission  line  when  D  is  the  spacing  of  the  wires  in  feet,  r  is  the  water  velocity  is  V  feet  per  second  through  the  tubes, 

the  radius  of  the  wire  in  inches  and  /  the    frequency.  L  is  in  feet  of  water  head. 

Take  /  as  60  cycles  or  some  desired  frequency  and  include  Problem  16. — Analyze    the    methods    of    construction 

a  scale  for  wire  sizes  in  connection  with  the  r  scale.  used  in  Figs.  37a,  d7b,  37c  and  37d. 


500 

1 

, 

B 

q 

?000 

-i 

^ciluesofT 

1            1.1 

53. 

' 

+ 

7,600° 

KkIO^ 

*\ 

■^ 

1 

M   M   M 

^ 

\^ 

^ 

Diaaram  frrequalion 

"i-AT^'e^T 
RicharJsons  Equation 
for  Thirm ionic  Current 
from  Heated  Metals  in  Vacua 
1  =  Current,amperes  cm'? 
T=  Absolute  Temperature 

For  pure  tvio,  b^S:36*IO''- 
..  ..  W,  b^-SSS.IO* 
"      "     Mo.  ^=1.1*10^ 

...      „      W.    h  =  22>cio-' 

— 

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fo  value  ofb,  ihenhorizonfallj/ 
io  value  of  A,  then  down 

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Amperes  per  cm  r 

-^ — liiijiifiilii  J  1 — 'tttt 

T*r+- 

r+n- 

-mT 

T+T+ 

U-J 

TTTT 

Ur+ 

't+T 

AH     1 
p^%iii,i,i, 

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T+T+ 

w 

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A'tZ'lo' 
A'lUlO^ 


iOlO      .  1.00 »  05     0.?   0.2 

+0.S  0 


Values  of  I 

Fig.  yrii. 


CHAPTER  III 


ALIGNMENT  DIAGRAMS  OR  COLLINEAR  NOMOGRAMS* 


11.  General  Tjrpe  of  Eqixation  and  Method  of  Treat- 
ment.^— There  will  be  considered  in  this  chapter  a 
great  class  of  formulas  which  may  be  written  in  the 
determinant  form 


h        Si         1 

h  g2  1 

/a         g3         1 


(8) 


in  which  /i  and  gi  are  functions  of  Zi,  f2  and  g2  func- 
tions of  22,  etc.  Such  knowledge  of  the  elementary 
properties  of  determinants  of  the  third  order  as  may 
be  gained  from  the  reading  of  Appendix  A  will  be 
assumed. 

A  distinguishing  characteristic  of  the  determinant 
of  Equation  (8)  is  the  presence  of  the  same  variable 
in  the  elements  of  each  row.  There  are  many  prac- 
tical formulas  which  may  be  reduced  to  this  form  and 
their  diagrammatic  representation  is  of  much  value. 
Such  formulas  lead  to  a  new  form  of  diagram  which 
will  be  called  the  alignment  diagram  because  its  key 
is  the  alignment  of  three  points. 

It  is  proved  in  analytic  geometry  that  if  three  points 
Pi,  Pi,  Ps  with  the  coordinates  (xiyi),  (xiyi),  (xzys), 
respectively,  lie  on  a  straight  line  (are  collinear)  the 
coordinates  satisfy  the  relation 


yi  1 
yi  1 
ya       1 


Xiy2  +  Xiys  +  xzyi  —  Xzyi 

—  Xtyx  —  ^iV3  =  0 


which  expresses  the  fact  that  the  point  P^ixiy-^  lies 
on  the  line  joining  the  points  P\{x\y^  and  P-iixiy-^  and 
whose  equation  is 


^  yt  -  yi 

Xi   -  Xx 


The  problem  is  then  to  establish  a  relation  between 
the  variables  Zi,  z-i,  Zz  of  Equation  (8)  and  the  position 
of  three  corresponding  and  inscribed  variable  points 
in  the  plane  such  that  whenever  three  values  of  z  are 
solutions  of  Equation  (8)  there  shall  correspond  three 
such  points  in  a  straight  line.  When  this  relation  is 
established,  a  straight  edge  applied  through  two  points 
'  Appendix  A  should  be  read  before  this  chapter. 


marked  with  known  values  of  z.-  Zj  must  pass  through 
one  or  more  points  marked  with  the  value  of  z*  which 
satisfies  Equation  (8). 

This  problem  is  most  easily  solved  by  using  the 
parametric  form  of  the  equations  of  plane  curves 
where  z  is  the  parameter.     The  equations 


X  =  /(z) 


g(2) 


are  the  parametric  equations  of  a  plane  curve  C.  For 
every  value  of  the  parameter  z  they  determine  a  point 
P  on  that  curve.  Three  such  sets  of  equations  will 
likewise  determine  three  curves  and  the  forms  of  the 
curves  will  depend  on  the  nature  of  the  functions  / 
and  g. 

Three  sets  of  such  parametric  functions  may  always 
be  determined  directly  from  Equation  (8).  If  the 
three  pairs  of  equations 


Xl  =/l 
X2    =  fi 

Xs  =  Ji 


yi  =  gi 

3'2    =    g2 

yz  =  g3 


are  formed,  using  the  elements  of  the  determinant  of 
Equation  (8)  in  the  order  shown,  they  may  be  con- 


sidered  as  the  parametric  equations  of  three  plane 
curves  Ci,  C2,  C3.  These  equations  will  be  called  the 
defining  equations.  When  the  curves  are  plotted, 
points  are  inscribed  with  corresponding  values  of  z 
and  thus  three  curved  function  scales  are  obtained. 
There  is  then  established  a  direct  correspondence  be- 
tween values  of  z  and  points  P  on  the  plane  curves  C. 


36 


DESIGN  OF  DIAGRAMS  FOR  EXGIXEERIXG  FORMULAS 


(See  Fig.  38.)     It  is  seen  therefore  that  if  Xiyi{i  =  1, 
2,  3)  in  the  equation 


a:i 

yi 

1 

X^ 

y-i 

1 

Xi 

3-3 

1 

are  the  coordinates  of  the  points  of  the  curves  defined 
by  the  three  pairs  of  equations  above,  then  Equation 
(8)  is  always  satisfied  by  values  of  Z\,  Zi,  23,  which 
determine  coUinear  points. 

When  an  engineering  formula  or  equation  in  three 
variables  is  given  for  which  a  diagram  is  desired,  the 
first  step  is  to  write  it  in  the  determinant  form. 
Equation  (8)  is  the  general  type  equation  in  three 
variables  for  which  corresponds  a  collinear  nomogram. 
Usually,  however,  an  equation  or  formula  does  not 
present  itself  in  a  determinant  form  nor  especially  in 
this  rather  simple  determinant  form.  Since  it  is 
always  necessary  to  establish  the  defining  equations 
before  constructing  a  diagram  it  is  very  desirable  to 
become  familiar  with  the  necessary  determinant  nota- 
tion at  once.  Equation  (8)  with  all  the  elements  of 
the  last  column  unity  is  called  the  reduced  determinant 
form.  It  is  almost  always  necessary  to  establish  a 
first  determinant  form  for  any  given  equation  and  then 
transform  it  by  the  laws  of  determinants  into  the 
desired  form  above. 

There  is  no  general  method  by  which  any  equation 
of  the  form 

/(21Z223)  =  0 
may  be  given  a  first  determinant  form  and  in  fact  not 
all  equations  in  three  variables  may  be  written  in  that 
determinant  form. 

Special  cases  of  Equation  (8)  have  been  studied  and 
the  necessary  and  sufi&cient  conditions  developed  for 
identifying  a  given  equation  with  them.  The  work 
involves  partial  differentiation  and  is  not  usually 
needed  in  practice.' 

12.  Diagrams  with  Three  Parallel  Straight  Scales. 
In  the  expanded  form  of  Equation  (8)  which  is 

fig2  +  figs  +  figi  ~  figi  -  fzgi  -  flg3  =  0      (9) 

should  ane  or  more  of  the  functions  /  or  g  reduce  to 
a  constant  and  especially  to  zero  the  equation  becomes 
much  simplified.     For  example,  the  equation 

/l+/2+/3  =   0  (6) 

previously  discussed  in  Chapter  II,  Article  9,  results  if 
gi  =  -1,  g2  =  1,  gs  =  0,  and /a  =  -  g 

>  Clark,  J.,  TWorie  Gfn^rale  des  Abaques  d'Alignment  de  toute 
Ordre,  Rivue  de  Micanique,  1907,  No.  39.  Also  d'Ocagne,  Nos.  152- 
153,  Traits  de  Nomographie. 


A  correspond 
(6)  is 

ng  first  determinant  form  of  Equation 
/:      -1          1     1 

h         1 

1 
1 

=  0 

Although  this  is  a  reduced  form  of  the  equation,  in 
the  sense  defined  above,  it  is  usual  to  write  this  equa- 
tion in  the  form  resulting  from  an  interchange  of  the 
first  two  columns  thus 


-1    /l    1 
1    fi     1 

0-§        . 

=  0                (10) 

The  defining  equations^  of  the  three  corresponding 
scales  are 

^=-1                        y=f. 

x=      \                          y=fi 

x=      0                           y=-^i 

and  the  scales  are  consequently  graduated  -on  three 
equidistant  parallel  lines.  This  is  perhaps  the  sim- 
plest form  of  collinear  nomogram  or  diagram  of 
alignment. 

Example  20. — By  the  method  of  "end  areas"  the 
volume  of  earthwork  per  station  on  railway  and  high- 
way construction  is  given  by  the  formula 

KV  =  {p,  +  Pi) 

where  V  =  volume  in  cubic  yards, 

A'  =  a  constant  depending  on  the  length  of  sec- 
tion and  scale, 
pi  and  p2  are  average  planimeter  readings  in 
square     inches     from     the     cross-section 
drawings. 
Comparing  this  formula  with  Equations  (6)  and  (10) 
it  is  seen  that  the  necessary  defining  equations  are 


-1 

y  =  pi 

1 

0 

y  =  pi 
KV 

The  diagram  (for  the  scale  of  cross-sections  4  feet  =  1 
inch)  may  be  constructed  with  the  vertical  unit  one- 
tenth  of  an  inch  and  the  horizontal  unit  5  inches.  If 
desired  the  scales  may  be  broken  and  repeated  to 
avoid   unduly  enlarging  the  diagram.     See  Fig.  39. 

It  usually  happens  that  for  the  range  of  values  of 
the  variables  involved  in  the  Equation  (6)  it  is  neces- 

'  Henceforth  it  will  be  sufficiently  clear  that  three  curves  are  under 
consideration  without  using  subscripts  to  distinguish  the  coordinates 
of  their  respective  points. 


ALIGNMENT  DIAGRAMS  OR  COLLI  NEAR  NOMOGRAMS 

IT- 1000 
950 


8  — 


7.11 


-100 

E-iso 


-II 


^ 


37 


Fig.  39.-Diagram  for  KV  =  [p,  +  p.)  V  =  Volume  of  Earthwork,  cu.  yds.  p,  and  p,  =  Average  Planimeter  Readings,  sq.  in 


38 


DESIGX  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


sary  to  introduce  scale  factors  and  sometimes  it  is 
desirable  to  establish  the  scales  at  unequal  distances. 
Suppose  that  it  is  desired  to  introduce  the  scale  factors 
ixi  and  Hi  on  the  parallel  scales  for  Si  and  s->  and  to 
estabUsh  these  scales  at  distances  5i  and  5..  from  the 
Y  axis.  It  is  then  necessary  to  determine  how  the 
third  scale  shall  be  graduated. 

The  new  defining  equations  for  the  first  two  scales 
would  necessarily  be  written 

X  =  -&i  y  =  Hifi 

X  =      Si  y  =  fi-f- 

and  it  may  be  assumed  temporarily  that  the  third 
scale  equations  will  have  the  form 

X  =  F3  y  =  G3 

where  7^3  and  G3  are  to  be  functions  of  /s  and  involve 
the  new  constants. 

To  determine  F3  and  G3  so  that  points  originally 
corresponding  to  any  set  of  solutions  of  Equation  (6) 
shall  remain  collinear  in  the  changed  diagram  it  is 
necessary  that  the  equation 

-  5  f^ifi         1 

62  M=/2  1 

F3        G3  1 

shall  be  satisfied  by  virtue  of  Equation  (6).     Upon 
expanding  this  equation  and  substituting  the  value  of 
/i  from  equation  (6)  there  results 
{t^i52  -  Mi^3)(/2  +/3)  -  {F3  +  5OM2/2  + 

(6i  +  52)G3  =  0 
Since  this  equation  must  hold  for  any  values  of  the 
independent  variables  S2  and  S3,  then  the  coefficient  of 
/2  and  the  term  not  involving  /«  must  vanish  identi- 
cally, that  is 

;il52   —    Ml^3   "~    M2^1    ~    ^2^3   =    0 

and  G3(5i  +  §2)  +  (mi52  -  miFs)^  =  0 

_   5i/X2    —   S-y/Xi  r,      —    —     ^'^'--^^ 

Ml  +  V-1 


whence 


G3 


+  M2 

and  the  defining  equations  of  the  third  scale  are 
_  hiy.\  —  Sifi2  _  _    ft  11^2 f 3 

Ml  +  AI2  All  +  M2 

It  is  seen  that  when  (§2^1  —  Si/i2)  =  0,  the  new 
scale  will  remain  on  the  Y  axis  and  the  constants  may 
usually  be  so  chosen  that  this  is  true.  It  is  to  be 
observed  also  that  the  scale  factor  of  the  third  scale  is 
independent  of  61  and  52-  Frequently  5i  and  &2  may 
be  chosen  equal  in  which  case  /xi  and  fj.2  must  also  be 
equal  if  the  third  scale  is  to  remain  on  the  Y  axis; 
that  is  to  say  if  the  three  scales  are  to  be  at  equal 

/UlM2 


distances.     The  quantity  - 


+   M2 


may  be  called  the 


As  a  check  on  the  work  the  values  of  F3  and  G3 
above  determined  may  be  substituted  in  the  last 
determinant  equation  with  the  result 

-Si  /Xl/l  1 

62  /.2^2  1 

52M1  —  S1JX2   —1x1112/3  J 


MlMiC^l  +  ^2) 

Ml  +  M2 

[/.  +  /2+/3]=0 


Ml  +  ^2        Ml  +  M2 

It  is  well  to  point  out  here  that  the  effect  of  the 
introduction  of  the  above  scale  factors  and  the  change 
of  moduli  is  to  apply  a  projective  transformation^  (see 
Appendix  B)  to  the  original  geometric  configuration. 
A  projective  transformation  when  applied  to  all  the 
variable  elements  of  the  first  two  columns  of  such  a 
third  order  (reduced)  determinant  has  the  effect  of 
manipulating  the  elements  of  the  determinant  by  the 
laws  of  determinants  and  the  net 'result  is  always 
merely  to  multiply  it  by  a  constant.  In  the  present 
case  the  constant  is 

_  MlM2(5i  +  S2) 
Ml  +  M2 

To  understand  how  the  above  theory  of  the  scale 
factors  is  applied,  the  formula  for  volume  by  "end 
areas"  of  Example  20  may  be  resumed.  The  use  of  a 
horizontal  unit  of  5  inches  and  a  vertical  unit  of  one- 
tenth  of  an  inch  was  equivalent  to  the  introduction 
of  the  values 

5i  =  So  =  5,  Ml  =  M2  =  ^io 
in  order  to  change  the  defining  equations  for  the  dia- 
gram to 


-5 


x  =  5 


x  =  0 


Pi 

y=ro 
P2 

KV 

y=^ 

It  is  to  be  observed  strictly  that  in  all  the  equations 
above /s  is  the  value  appearing  in  Equation  (6). 

By  using  a  logarithmic  transformation  any  equation 
of  the  form 

Z{'  =  KZ2W  (11) 

(a,  /3  and  y  =  constants)  maybe  written  in  the  form 
of  equation  (6)  thus 

a  log  zi  -  fi  log  S2  -  7  log  S3  -  log  A'  =  0 
The  corresponding  diagram  has  three  parallel  loga- 
rithmic scales  defined  by  the  equations 
X  =  —I  y  =  a  log  Si 

X  =       1  y  =  /3  log  S2 

x=      0  y  =  -J^(7logs3 -H  log  A') 

'  The  projective  transformation  has  the  equations 


(miSz  4-  M2ii)»  -I-  {f^i&2 


.«.) 


scale  factor  /is  of  the  third  parallel  scale. 


(in  -  tit)x  +  (mi  -t- 
2MiM2y 

-   /X2)X  +    U,  -I-  M2) 


■M2) 


ALIGNMENT  DIAGRAMS  OR  COLLINEAR  NOMOGRAMS 


39 


The  following  equation 

z'z-z^Zi''  =  constant  (12) 

may  be  similarly  treated.  The  logarithm  of  the  con- 
stant can  of  course  be  associated  with  any  one  of  the 
variables  desired  for  convenience  in  constructing  and 
using  the  diagram. 

Example  21. — An  illustration  of  Equation  (11)  is 
afforded  by  the  formula  for  the  volume  of  a  torus  or 
ring  of  circular  cross-section 

V  =  2AQ74:Dd^ 
Taking  logarithms  of  both  sides  of  this  equation  it 
may  be  written 

2\ogd  +  log  D  -  log  F  -I-  log  2.4674  =  0 
A  corresponding  reduced  determinant  form  is  therefore 
-1  •       2  1og^  1 

1  log  Z?  1 

log  2.4674  -  log  V 
0  2  1 

so  that  the  three  scales,  when  no  scale  factors  are 
used,  are  defined  as  follows: 

x=  -1  y  =  2\ogd 


x=      1 
x=      0 


Iog2.4674-logF 


If  the  same  limiting  values  are  chosen  for  d  and  D  it  is 
seen  that  the  scale  for  d  will  be  twice  as  long  as  that  for 
D.  In  order  to  have  these  scales  of  the  same  length 
and  covering  the  same  range  of  values  and  so  arranged 
that  both  may  be  read  with  equal  accuracy,  choose 

Ml  =  1  ^2  =  2 

For  convenience  let  {tnh  —  fii^i)  =  0  so  that  82  = 
2S1  and  the  scale  factor  for  the  V  scale  will  be 

MlA'2  2 

M3   =   X ^    ■? 

Ml  +  W         o 

The  constant  term  log  2.4674  in  the  V  function  simply 
determines  the  initial  point  of  the  logarithmic  scale 
for  V,  (see  Fig.  40) ,  for  which  the  equations  are 
x  =  0,  y  =  M[log  V  -  log  2.4674] 

It  frequently  happens  that  two  parallel  scales  will 
extend  in  opposite  directions  from  the  X  axis  and 
whenever  this  is  so  a  displacement  of  the  scales  along 
their  supports  is  desirable  in  order  to  dispose  them  to 
better  advantage  on  the  sheet.  In  the  following 
example  the  K  and  R  scales  are  started  from  a  line 
making  an  angle  of  45°  with  the  X  axis  at  the  initial 
point  of  the  A  scale  while  the  original  distance  between 
the  scales  is  preserved.  Geometrically  this  is  the 
effect  of  carrying  out  upon  the  original  diagram  a 
projective  transformation  whose  equations  are 

xi  =  X  yi  =  X  +  y  +  1 

and  consequently  alignment  is  preserved. 


Example  22. — The  area  of  a  segment  of  a  circle  of 
radius  R  and  height  H  is  given  by  the  exact  formula 
H 


H- 


A  =  RH  arc  vers  =-  - — (R  -  E) 

Since  H  appears  always  divided  by  R,  write  -^  = 
then 

A  =  R'[aTc  vers  A"  -  V2K  -  K-{1  -  K)] 
and  passing  to  logarithms 
log  yl  =  2  log  R  +  log  [arc  vers  A'  -  V2K  -  h 


so  that  the  reduced  determinant  form  is 
-1  log^ 


■2  log  7? 


(1  -  A)] 


0  Yi  log[arc  vers  A  -^2K  -  ^=(1  -  K)] 
Figure  -41  shows  the  diagram  for  this  formula  con- 
structed with  unit  scale  factors.  When  A'  =  1  and 
when  A  =  2  the  corresponding  areas  are  respectively 
semi-circles  and  circles. 

In  most  practical  examples  the  displacement  of  a 
scale  whose  graduations  increase  in  a  downward 
direction  from  the  X  axis  is  best  effected  as  in  this 
example  by  simply  starting  it  from  a  point  above  the 
X  axis  on  a  45°  line  through  the  origin. 

Example  23. — Figure  42  shows  a  diagram  for  the 
formula 

P  =  CF^D'^' 
which  is  given  by  F.  W.  Taylor'  for  the  pressure  on  a 
tool  when  cutting  cast  iron,  where 
F  =  feed  in  inches, 
P  =  pressure  in  pounds, 
C  =  45,000  for  soft  cast  iron, 
C  =  69,000  for  hard  cast  iron, 
D  =  depth  of  cut  in  inches. 
Passing  to  logarithms,  the  formula  becomes 

H  log  F  +  ms  log  D  =  logP-  log  C 
and  the  scales  are  defined  by  the  equations 
X  =       1  >'  =  ^•^  log  F 

a;  =  —  1  y  =  ^Hs  log  D 

x=      0  y  =  Viilog  P  -  log  C] 

The  constant  C  is  associated  with  the  P  scale  in  order 
that  its  extreme  values  given  above  may  be  used  in 
placing  the  graduations  on  the  P  scale.  The  diagram 
thus  gives  the  maximum  and  minimum  values  of  P 
for  any  D  and  F. 

Example  24. — In  correcting  a  barometer  reading 
at  a  temperature  /i  to  a  temperature  /  for  which  the 
barometer  is  calibrated  the  correct  reading  in  English 
units  would  be 

h  =  //,[1  -  0.000101(/i  -  t)] 
»  Trans.  A.  S.  M.  E.,  vol.  28. 


40 


10- 

9-1 

8 

7- 

64 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 

r3000 
2000 


1000 
E-900 

800 

100 
hWO 

500 
-400 

-300 

200 
150 


2- 


O 


100 
•90 
80 
•TO 
GO 
50 

•40 
[-10 


r20 


1-10 
9 
8 
7 

■G 
■5 

■4 
E-5 


Diagram  for 
fhe  Volume  of  a  Torus 

V-2.4G74d^D ' 


rlO 
9 

8 
-T 

^5 


-2 


!-• 


100  ■ 


ALIGNMENT  DIAGRAMS  OR  COLLI  NEAR    NOMOGRAMS 


41 


D/'agram  for  fhe 
ExacfArea  of  a  Ci'rcu/ar  Sepmenf 


A  =  fi^[arc  vers^~  IIMB(R.ff)] 


r-I.O 


1-1 

G 


4^ 


nor 

o 


-2.0 

•  1.5 
■  1.25 

-1.0 
■0.9 
•0.8 

hoi 

■CG 
■0.5 
■0.4 

1-O.J 


-0.2 


-I.S 


-2.0 


D 


-3.0 


1.0-1 
0.9- 
0.8- 
0.7- 
0.&- 
0.5- 

0.4- 


•  0.1 


F-4.0 

-5.0 

-&.0 

t-7.0 
-8.0 

9.0 


42 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


l.(W-1.0 

.90 


54 


-V« 


.35- 

-»/.. 

05 

5         ^ 

-"/„ 

■S 

-v„ 

S-: 

-"/„ 

o        _ 

5    - 

-Vm 

p. 

o         — 

Q        J 

-"/„ 

JB^ 

-4 

I 

-'Vm 

T 

-V,» 

.20- 

-"/„ 

.19- 

-v.. 

10.000- 
9000- 
8000- 
7000- 


-§-10.000 
9000 


7000 
6000 


p 

; 

r 

n 

o 

a 

u 

s 

2000- 

-  3000 

1 

5 

1 

1 

"S 

1500- 

M 

w 

- 

J3 

:S 

1  2000 

'% 

& 

. 

1 

; 

• 

h 

i 

1000- 

-   1500 

a 

g 

90O- 

- 

o 

1 

800- 

- 

\ 

Fig.  42.— Diagram  for  P=  CF^  T^K 


ALIGNMENT  DIAGRAMS  OR  COLLI  NEAR  NOMOGRAMS 


43 


where  hi  is  the  observed  height.     The  correction  there- 
fore is 

E  =  h\  O.OOOIOIA/] 
or,  passing  to  logarithms, 

log  E  -  log  0.000101  =  log  hi  +  log  M 
If 

X  =  -bi  y  =  ixi\og  hi 

X  —       h  y  =  fi-i  log  A/ 

it  would  be  desirable  to  apply  the  methods  of  Article 

4  to  determine  the  scale  factors  /Ji  and  ;ii2  in  order  to 

extend  the  scale  resulting  from  the  short  range  of 

numbers  for  hi.     If  the  hi  and  At  scales  are  each  to  be 

10  inches  long  and  the  following  limits  chosen: 

hi  from  26  to  31  inches  of  Mercury 

A^  from  1  to  70  degrees  Fahrenheit, 

then 

Ml  =  130     and    ^2  =  5.42 
Taking  (/ii52  —  fi2&i)  =  0  as  before,  there  results 
6,  _  Ml  _  130       24 
6o  ~  M2  ~  5.42  ~    1 
and  for  m3 

5.19 


Ml  +  M2 
The  defining  equations  then  are 

x=  -24  y  =  130  log /fi 

X  =        1  y  =  5.42  log  At 

x=        0  y  =  5.19  [log  £-  log  0.000101] 

and  the  diagram  appears  in  Fig.  43. 

Example  25. — A  modification  by  Grashof  of  Napier's 
Rule  for  the  flow  of  steam  through  an  orifice  is  some- 
times used  for  steam  nozzles  in  the  following  form 
„       P'-"Ao 
^  ~      60 
where 

F  =  flow  of  steam  in  pounds  per  second, 
P  =  absolute  initial  pressure  in  pounds  per  square 
inch, 

A  0  =  area  at  throat  in  square  inches. 
If  T/ritten  in  the  logarithmic  form,  then 

log  F  +  log  60  =  0.97  log  P  +  log  ^0 
In  order  to  use  the  same  units  for  the  F  and  the  ^o 
scales  let 

^'"~  =  nn7  Ml  =  1 


0.97 

then  if 

^2    _    M2    _    _1_ 

6i  "  Ml  ~  0.9 
the  three  scales  are 

X  =      0.97 

x=  -I 

x=      0 
See  Fig.  44. 


1 
1.97 


0.508 


y  =  logylo 

y  =  log  P 

y  =  0.508[log  F  +  log  60] 


Example  26. — The  Royal  Automobile  Club  (Eng- 
land) automobile  engine  rating  gives  the  rated  horse- 
power, HP,  of  N  cylinders  of  bore  D  inches  and 
stroke  5  inches  as 

D'-NS 
12 


HP 


or 


m 


+  log  12  =  2  log  Z)  + 


In  order  to  read  the  D  and  5  scales  on  a  diagram  with 
equal  ease,  let 

X  =  —di  y  =  Ml  2  log  Z> 

X  =         &2  y  =   M2  log  5 

HP 


0 


log 


N 


log  12 


if      Ml  =  K,     M2  =  1     then  ms  =  M-     See  Fig.  45. 

It  is  to  be  observed  that  when  it  is  desirable  to 
displace  one  or  more  of  the  parallel  scales  in  a  diagram 
it  is  not  necessary  to  start  the  downward  scales 
from  a  Une  making  45°  with  the  X  axis  but  any  angle 
a  whatever  may  be  used.  The  equivalent  projective 
transformation  in  the  case  of  a  diagram  with  scales 
originally  at  distances  Si,  5o  from  the  Y  axis  would 
have  the  equations 

Xi  =  X  yi  —  {x  -\-  5i)  tan  a  -\-  y 

Equations  of  four  variables  of  the  form 

/l+/2+/3+/4   =    0  (13) 

may  be  represented  by  parallel  scale  diagrams  and 
will  be  discussed  in  Chapter  IV  together  with  the 
more  general  type 

/1+/2+/3+/4+    .    .    .   +/„  =  0 
13.  Diagrams  with  Straight  Scales  and  Two  Only 
Parallel. — It  is  easily  seen  that  the  equations 

x  =  0  y  =  gi 

x=\  y=  go 

x=fz  y  =  0 

where  gi,  gi,  fs  are  functions  of  Zi,  Zi,  Zi  respectively, 
would  define  a  diagram  in  which  there  would  be  two 
parallel  scales  and  a  third  straight  scale  perpendicular 
to  them.  What  is  the  corresponding  equation  in 
three  variables  for  which  such  a  diagram  would  be 
useful? 

Before  deciding  this  question  it  is  well  to  state  that 
all  equations  or  formulas  are  subject  to  a  great  variety 
of  algebraic  and  other  transformations:  clearing  of 
fractions,  factoring,  removal  of  radicals,  separation 
or  combination  of  constants,  etc.,  which  all  tend  to 
change  the  appearance  of  any  given  equation. 

An  equation  corresponding  to  the  particular  type  ■ 
of  coUinear  nomogram  or  diagram  of  alignment  to  be 
discussed  is  not  diflacult  to  establish  for  it  is  only 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


31.0- 


0.2190-1    ^10 
0.?000— I     I    ,^ 

50 
^40 


30.0  — 


^    ?9.0 


2S.0- 


27.0 


2G.0— 1 


Diagram  for  Correcfing 
Readings  of  Mercury  Baromehsrs 

from  formula  h=h,[l-  0. 000 1 01  (i,  -  ffj 

h-  Correcied  heighi  of  column 
h,  =  Observed  heic/hf  of  column 
fj  =  Observed  iemperafure 
"f  -  Calihrafion  iemperalure 


-  -  '  -OTiobo 

0.0800- 
0.0600^ 


?  0.0400- 


0.0200- 


0.0100- 


50— 


0.0060— 


0.0040-5 

0.0050-^ 
0.002G— 


—30 


■20 


-10[| 

•9      1/5 


■5 


—4 


—3 


-7         ^ 

6- 


L, 


50- 

4-0  4 

30 

20- 


10- 


7  — 
5-= 


0.5- 


ALIGNMENT  DIAGRAMS  OR  COLLI  NEAR  NOMOGRAMS 

500- 

200- 
150- 

100; 

GO- 

50- 

40- 

50 
20- 

10^ 
I 

7 
<o—\ 

5 
4 

l- 
2  — 


1.0- 

Si 

0.7  ■ 

0.6 

0.5- 

0.4. 

0.3 


0.2-= 


0.1- 


0.05- 


^ 0,97- 


0.02-1 

Fig.  44. 


Diagram 
for  fhQ  Formula 

p.  Aopr' 

GO 


1.00 


^ 


45 

•550 
-500 
■250 

-ZOO 
-150 


■100 
-90 
-80 
-70 
-GO 
-50 
1-40 

E-30 
-20 


10 

r^ 

-8 
-7 

-5 


-3 


46 


G.OO. 

5.1S- 

5.50- 

5.25- 

5.00- 

4.75- 

ft) 

I  4.50 


ft? 


.4.25- 


o 

u 

-fe  4.00H 

E 
o 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 

22 

eo 

F-18 

n 

IG 
-15 

-\A 
\l 


3.75- 


3.50- 


3.25- 


J.00-1 


-7 


-G 


-5 


-4. 


o 


D/AGffAM 
For  the  Formula 

English  Club  Auhmobile 
Rating:-  To  Find  iohl 
horsepowerjmulFpIg  ihe 
horsepower  per  cylinder 
bi/  number  ofcj/lihders. 


■8.00 
7.15 
T50 
■7.25 
-7.00 
•G.75 
■G.BO 
-G.25 

•4-G.OO 


5.75  J 
o 


5.50'- 
5.25^ 
-5.00 

-4.7S 


-4.50 


4.25 


•4.00 


ALIGNMENT  DIAGRAMS  OR  COLLI  NEAR  NOMOGRAMS 


47 


necessary  to  write  the  reduced  determinant  equation 
corresponding  to  the  above  defining  equations 

0  gi         1 

1  g,        1=0  (14) 
/3          0            1 

But  this  reduced  determinant  equation  may  have 
resulted  from  any  one  of  an  indefinite  number  of  first 
determinant  forms  all  of  which  are  equivalent  under 
the  laws  of  writing  determinants  without  changing 
their  values  and  the  corresponding  diagrams  would 
either  be  alike  or  equivalent  to  each  other  by  a  pro- 
jective transformation. 

The  expanded  form  of  the  determinant    equation 
above  is 

/sfg.  -  g-2)  -  g.  =  0  (15) 

but  a  more  simple  form  of  the  expanded  equation 
which  would  yield  exactly  the  same  type  of  diagram 
would  be 

//-/,V  =  0  (16) 

and  the  two  equations  are  equivalent  in  view  of  the 
following  relations 

The  corresponding  simple  first  determinant  form  for 
Equation  (16)  is 

/:'  1  0 

-N      0      1 
0         \       h 

and  the  reduced  determinant  equation  is  obtained  by 
adding  column  two  to  column  three  to  form  anew  third 
column,  then  dividing  the  elements  of  the  third  row 
by  (1  +  liz)  and  then  interchanging  columns  one  and 


U  =  g.-U 


two   and   rows  one  and  two. 
minant  equation  is 

0  -// 

1  /i' 
1 


The  resulting  deter- 


1+. 


0 


(17) 


and  it  is  only  necessary  to  inspect  the  relations  written 
above  to  identify  this  equation  with  Equation  (14) 
which  will  henceforth  be  considered  the  reduced  deter- 
minant form  for  an  equation  whose  diagram  has  two 
parallel  scales  and  a  third  perpendicular  scale.  On  the 
other  hand  Equation  (16)  will  be  the  type  form  for  the 
expanded  equation  and  Equation  (17)  is  very  useful  to 
determine  the  defining  scale  equations.  No  determi- 
nant form  of  any  equation  should  ever  be  used  with- 
out first  expanding  it  to  check  the  correctness  of  the 
determinant.  It  is  to  be  noticed  that  by  transposing 
and  passing  to  logarithms  Equation  (16)  becomes 
identical  in  form  with  Equation  (6)  of  Chapter  II. 


It  is  usually  necessary  to  introduce  scale  factors  for 
the  construction  of  the  parallel  scales  of  Equation(14) 
and  to  move  one  scale  a  distance  5  from  the  other. 
Let  the  corresponding  defining  equations  for  the 
changed  parallel  scales  be 

X  =  0  y  =  tngi 

X  =  b  y  =  ,jL2g2 

To  determine  the  third  scale  equations  one  may  pro- 
ceed as  in  Article  12.  Assume  that  the  defining  equa- 
tions will  have  the  form 

X  =  Fz  y  =  G3 

where  F3  and  Gz  are  functions  of /a  and  involve  m,  H2, 
and  5.  Then  the  changed  determinant  equation  from 
(14)  will  be 


/iigi 

M2g2 

Gz 


/Ulgl/^3  +    bGz 


gl 


M2g2^3   - 

SMlgl  =  0 
But  from  the  expanded  form  of  Equation  (14) 

g2/, 
73-1 

and  upon  substituting  this  value  of  gi  it  is  necessary 
that  the  previous  equation  become  an  identity  for 
every  value  of  g2.  Hence  equating  the  coefficient 
of  g2  and  the  term  not  involving  go  to  zero  there 
follows 

p ^^'/3  r    -  r\ 

rz  —  7  NT     ,  Cj3  —  U 

and  the  third  pair  of  defining  equations  becomes 

5M./3  „ 

X  =  -. s  f     , —  y  =  0. 

IMI     —    fil    /3    -|-    M2 

It  is  seen  that  the  scale  on  the  X  axis  may  be  if 
desired  obtained  from  a  scale  of  the  function /a  by  the 
methods  of  Chapter  I,  Arricle  2{d). 
Example  27. — The  formula  of  Francis 
Q  =  S.^SBH^' 
may  also  be  given  the  determinant  form 


0          -Q 

1 

1         3.335 

1 

=  0 

H^l     « 

1 

id  the  defining  equations  are 

x  =  0 

X  =  8 

y  =  M23.335 

dn.H^i 

y=  0 

"^  -    (mi    -  M2)i?^^  +  M2(H^^ 

+  1) 

id  if  5  =  1  there  results  for  the  H  scale  simply 

X 

~    f^^H^'    +  M2 

y  =  0 

48 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


ALIGNMENT  DIAGRAMS  OR  COLLINEAR  NOMOGRAMS 


The  corresponding  diagram  is  shown  in  Fig.  46  and 
the  scales  have  been  displaced  by  the  method  of  the 
preceding  section. 

Example  28. — The  formula 

which  gives  the  natural  draft  D  in  inches  of  water 
available  from  a  chimney  H  feet  high  when  the  tem- 
perature of  the  gases  is  /  degrees  Fahrenheit,  may  be 
written 


[0.014671  -4^^f-^ 


1  -D 

FTfl  " 

In  Fig.  47  the  diagram  is  shown  with 

Ml  =  833  112  =  4 

and  the  scales  are 


X  =  0 

a;=  1 


833  0.014671- 


1 
1 

=  9.16 

7.644 

461  +  / 


(9.16)(833) 


1  +  H 


(833 


^hi-H+^ 


7.630 
833  +  4H 


The  values  of  ^i,  M2  and  5  were  chosen  in  accordance 
with  the  methods  of  Article  4. 

In  Equation  (8)  of  Article  11  if  any  pair  of  the 
functions  /<  and  g.-  should  have  the  form 
aiZi  +  bi  a^Zi  +  h 


a^Zi    

OsZi  +  bi 


1,  2,  3 


oz^i  +  bi 

then  the  corresponding  defining  equations  would  still 
determine  a  straight  scale,  for  upon  elimination  of  s,- 
from  a  pair  of  such  defining  equations,  there  would 
result  the  straight  line  equation 

(0263  —  a3b2)x  +  (^361  —  aib3)y  —  (ao&i  —  ai&2)  =  0 
In  case  all  the  functions  in  Equation  (8)  had  the  above 
form  all  three  scales  would  be  graduated  on  straight 
lines  and  no  two  would  in  general  be  parallel.  All 
the  equations  so  far  discussed  in  this  chapter  are 
special  cases  of  such  forms.  No  essentially  new  type 
of  diagram  would  result,  however,  in  the  more  general 
case,  for  a  projective  transformation  could  always  be 
found  to  change  any  such  set  of  three  given  lines 
into  a  corresponding  set  either  of  all  parallel  lines  or 
of  two  parallel  and  one  non-parallel.  For  any  set  of 
three  lines,  no  two  of  which  are  parallel,  intersect 
either  in  one  or  in  three  points  and  it  is  only  necessary 
to  apply  such  a  projective  transformation  to  the 
configuration  as  will  transform  a  point  of  intersection 


P  with  the  coordinates  {m,  n)  to  infinity;  then  either 
the  three  or  the  two  lines  formerly  intersecting  at  the 
point  become  parallel  lines  in  the  transformed  posi- 
tion and  one  or  the  other  of  the  cases  already  discussed 
would  result. 

To  require  that  a  point  P{m,  n)  be  transformed  to 
infinity  it  is  only  necessary  that  the  general  projec- 
tive transformation  whose  equations  are 


xi\-- 


Axx^  Biy  +  C\ 
A3X  +  Bay  +  C3 


yi 


Aix  +  BiV  +  Co 
A3X  +  Bay  +  Cz 


be  chosen  with  coefficients  Az,  B3,  C3  satisfying  the 
relation 

Aafn  -\-  Bzn  +  C3  =  0 

(See  Appendix  B.)  This  projective  transformation 
may  in  other  respects  be  selected  at  will  and  would 
of  course  be  made  as  simple  as  possible.  As  it  has 
already  been  stated  in  Article  12,  the  effect  of  apply- 
ing the  projective  transformation  to  the  elements  of 
the  original  determinant  equation  is  to  rearrange 
them  by  the  laws  of  determinants. 

In  practice  the  general  case  of  three  non-parallel 
straight  lines  seldom  occurs.  When  an  equation 
does  arise  with  the  determinant  form  above  dis- 
cussed it  is  of  course  not  necessary  to  transform  it  to 
a  form  with  parallel  scales  as  a  few  changes  in  the 
determinant  will  usually  simplify  the  work  of  plot- 
ting the  given  scales. 

14.  Diagrams  of  Alignment  with  Ctirved  Scales. — 
Before  discussing  equations  or  formulas  in  three 
variables  which  can  be  solved  with  diagrams  of  one 
or  more  curved  scales,  consider  the  general  quad- 
ratic equation 

z^  +  pz  +  q  =  0 

where  the  variables  are  z,  p,  and  q.  To  set  up  the 
defining  equations  of  a  diagram,  the  equation  may  be 
reduced  to  a  suitable  determinant  form,  bearing  in 
mind  that  no  row  shall  contain  more  than  one  of 
the  variables. 

Select  for  the  first  row  functions  /i  =  z,  and  gi  = 
z^  and  leave  the  third  element  position  blank. 

Then  select  g2  =  p  to  combine  with  z  and  leave 
the  other  positions  of  the  second  row  blank;  there 
results 


Now  insert  q  as  gs  to  avoid  combinations  with  2 
and  leave  the  remaining  positions  blank  thus: 


50 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


Then  since  the  elements  z  and  q  and  the  product 
pz  are  terms  of  the  original  equation,  there  is  sufficient 
guide  to  complete  the  determinant  as  follows: 

2     -z'         1 

1        p         0 

0  q  1 


750 -, 


corresponding  rows  divided  by  its  elements.  But  this 
would  require  the  use  of  the  reciprocals  of  p  and  q. 
To  avoid  using  the  reciprocals  add  the  corresponding 
elements  of  columns  one  and  three  and  form  a  new 
third  column.  Dividing  the  elements  of  the  first 
row  by  (1  +  z)  there  results  finally: 


0.0 
0.1 
0.2 
0.3 
0.4 
0.5 
O.G 
•0.7 

-0.8 
-0.9 
-1.0 


Diagram  for  Chimnet/  Formula 
D-(0.S2)(l4-VH[^,-j] 


To  put  this  first  determinant  equation  into  the 
reduced  form  (8),  there  are  available  all  the  elementary 
laws  of  determinants,  and  changes  in  sign  of  the 
determinant  may  be  disregarded  (not,  however, 
changes  in  sign  of  the  various  elements  unless  suffi- 
cient to  change  the  sign  of  the  determinant).  Since 
the  second  column  is  free  from  zero  elements  it  might 
be  moved  into  the  third  column  position  and  then  the 


z  — z- 

1 +z  1 +z 

1         p 

0  q 

Comparing  this  determinant  with  Equation  (8)  the 
equation  of  the  three  scales  may  for  convenience  be 
written  as  follows: 


ALIGNMENT  DIAGRAMS  OR  COLLI  NEAR  NOMOGRAMS 


51 


Example  29. 


X  =  -1 

.T   =         0 


'  1+2 

y=    P 
y  =    q 


The  scale  for  s  is  then  graduated  on  an  hyperbola  with 
the  asymptote  x  =  —I.  Figure  48  is  the  diagram. 
No  scale  factors  are  used  but  the  unit  on  the  horizontal 
axis  is  taken  100  times  that  on  the  vertical  a.xis,  as  this 


^70 
60   a. 

50  i 


Diagram  for  fhe  Quadratic  '^^' 


rlOO 

90 

80 

10 
L60 

■50  ^ 

40 1 

30 

20 


convenient  device  is  always  available.  The  roots 
are  read  at  the  points  where  the  straight  line  through 
the  given  values  of  p  a-nd  q  on  their  respective  scales, 
cuts  this  hyperbola  graduated  with  s. 

Consider  now  the  process  of  obtaining  a  first 
determinant  form  for  any  given  equation.  The 
following  steps  outline  a  method  of  trial  and  error 
which  will  reduce  most  of  the  formulas  of  engineering : 

First. — Select  three  or  less  functions  of  one  variable 
which  will  deplete  the  formula  of  that  variable  and 
arrange  them  in  any  order  as  elements  of  a  first  row, 
leaving  missing  elements  blank. 


Second. — Arrange  similar  functions  of  a  second 
variable  as  elements  in  a  second  row  so  that  products 
required  by  the  formula  will  result. 

Third. — The  remaining  functions  are  elements  of  a 
third  row  and  are  inserted  with  regard  to  the  resulting 
combinations. 

Fourth. — Supply  by  inspection  necessary  constants 
(including  zero)  for  missing  elements,  and  rearrange 
terms  in  the  rows  until  the  expanded  determinants 
check  with  the  forms  of  the  given  equation. 

When  a  first  determinant  is  found  it  can  readily  be 
transformed  to  the  type  (8)  by  the  laws  of  determi- 
nants.    (See  Appendix  A.) 

Example  30. — In  Fig.  49  is  shown  a  diagram  for  the 
cubic  equation 

2'  +  /.Z  +  g  =    0 

A  first  determinant  form  of  the  equation  may  be  found 
by  replacing  —  s^  by  —z^  in  the  first  determinant  form 
for  the  quadratic  equation.     The  reduced  determinant 
form  used  for  the  present  diagram  was,  however, 
;-  1  -z' 

:+l  2+1 

1  p  1 

-1  q  1 

The  corresponding  scale  equations  are 

■=  -~  ^  =    -2^ 

X  =  \  y  —  p 

x=  -1  y  =  q 

A  simple  form  for  the  reduced  determinant  equation 
to  which  will  correspond  a  diagram  with  two  parallel 
straight  scales  and  a  curved  scale  is 

1  gi         1     I 

0        g-i        1=0  (18) 

h       g3        1 
The  quadratic  equation  above  is  an  example.     When 
this  Equation  (18)  is  expanded  there  results  the  form 
g2+Mgi-g2)-g3  =  0  (19) 

The  functions  /a  and  gs  must  of  course  not  have  the 
linear  form  discussed  in  the  preceding  section.  The 
procedure  for  the  introduction  of  scale  factors  in 
constructing  corresponding  scales  is  exactly  as  in  the 
case  of  the  equation  of  Article  13.  If  the  first  two 
defining  equations  are  written 

X  =  8  y  =  ^igi 

x=  0  y  =  M2g2 

then  the  third  pair  of  equations  for  the  new  curved 
scale  will  have  the  form 

x  =  Fs  y  =  Gs 

where  F3  and  G3  will  be  functions  of  fs  and  gs  and 
involve  the  new  constants  6,  m  and  H2-     Upon  sub- 


62 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


ro  W 


o         o         o 


^^s^ 


<o  0) 


^O  53.1]  DA 


ALIGNMENT  DIAGRAMS  OR  COLLI  NEAR  NOMOGRAMS 


53 


stituting  the  value  of  gi  from  Equation  (19)  into  the 
equation 

0  ti-igi  1  =    6M2g2  +   Ml^3gl   -    IJ-tglFi 

F3      G,         1     1  -6G3=0 

there  results  the  equation 


g2     6M2   +   Ml^3— 7 A'2i^3  1    H 


6G3  =  0 


.13  J  /3 

which  must  be  true  for  every  go,  hence  the  coefficient 
of  go  and  the  term  not  involving  g-.  must  vanish  iden- 
tically and  there  follows,  upon  equating  these  expres- 
sions to  zero 

^'  -  M2/3  -  Mi(/3  -  D'  ^'  ~  ,2fz  -  Mi(/3  -  1)  ^^""^ 
The  above  equations  result  also  from  the  application 
to  the  original  figure  of  the  following  projective 
transformation 

gM2x MiM2y  .^.s 

^'  -  {y.,  -  M.)X  +  Ml'  ^'  ~  U  -  Mi)x  +  Ml  ^  ^ 
and  this  projective  transformation  may  be  obtained 
by  the  methods  explained  in  Appendix  B. 

There  are  given  below  two  examples  of  equations 
arising  in  surveying  practice  for  which  diagrams  are 
very  useful  and  in  both  of  which  curved  scales  occur: 

Example  31. — Stadia  Formula  for  Horizontal  Dis- 
tance.— The  distance  H  oi  a  point  from  the  instrument 
is 

H  =  R  —  R  sin^  a  +  c  cos  a 

where:  H  =  horizontal  distance  in  feet 

R  =  rod  reading  multiplied  by  100 
a  =  vertical  angle 

c  =  instrument  constant,  0.85  to  1.15 
With  c  taken  as  unity  (which  is  sufficiently  exact  for 
most  work)  the  formula  may  be  given  the  first  deter- 
minant form 

\     I         H  1 

0         i?  1 

I  1  cos  a  sin^  Q 
When  this  determinant  equation  is  given  the  form  (8) 
by  division  of  the  last  row  by  sin^  aand  the  correspond- 
ing defining  equations  established,  it  is  seen  that 
because  of  the  small  values  of  the  angle  a  (a  seldom 
exceeds  30°)  the  diagram  is  impracticable.  However, 
by  adding  the  first  and  last  columns  to  form  a  new 
third  column  and  then  dividing  by  the  elements  of 
the  new  third  column,  there  results  the  form 

^  ^  1 

2  2  ^ 

0  R  1 

1  cos  a 


1  +  sin'-a    1  +  sin- 


/hich  may  finally  be  arranged  as  follows 
1  H  1 

0  2R  1 

2 2  cos  a 

1  +  sin- a    1+  sin' a 

In  this  form  which  corresponds  to  Equation  (18)  the 

defining     equations,    with    the    needed  scale  factors 

1  1  1 


"  -  21   ^'  -  1,050 
Equations  (21) 
1 
*  ~  21 


2,000 


0 


may  be  written  from 

H 
1,050 

R 
1,000 

1  cos  a 

^  "  1+20  sin^  a  ^'  ^  50[1  +  20  sin^  a] 
With  the  modulus  unity  equal  to  20  inches  the  diagram 
of  Fig.  50  was  constructed.  The  scale  for  a  is  gradu- 
ated on  an  hyperbola  tangent  at  its  vertex  to  the  line 
X  =  yix-  When  a  =  0  the  corresponding  points  on 
the  scale  are  x  =  1,  y  =  }4o-  In  order  to  bring  small 
values  of  the  angle  a  within  the  limits  of  the  drawing 
the  transformation 

xi  =  X  yi  =  x  +  y 

was  applied  to  the  first  diagram  as  defined.  For 
actual  practice  in  the  office  or  field  more  suitable 
values  of  the  constants  may  doubtless  be  chosen. 

Example  32. — The  formula  for  the  vertical  distance 
of  a  point  above  the  level  of  the  instrument  is 

and  this  formula  may  be  given  the  first  determinant 
form 

1  V  0 

0  R  1 

sin  2a 
-1  -sma        -^— 

By  suitable  transformations  there  results  from  this 
form  the  reduced  determinant  form  of  the  equation 
1  V  1 

0  R  1 

-2  -2  sing 

sin  2a-  2     sin  2a  -  2 
using    Equations    (21)    and   the  scale  factors 


sin  2a  -f  sin  a 


Again 
2 
S 


_  _      25 

21'    *"  ~  10,000'  ^- 
defining  equations 
2 


-r-x7j7r>  there  result  the 


2.5  sin  2a 


54 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


R      ..H        V 


\ 


fi>=  70O5«L 


--?7''30_ 


\ 


\ 


\ 


\. 


Fig.  50.— Diagram  for  the  Stadia  Formulas  F  =  /?-/?  sin' a  +  cos  a  and  K 


ALIGNMENT  DIAGRAMS  OR  COLLINEAR  NOMOGRAMS 


55 


The  values  of  a  are  graduated  on  a  quartic  curve. 
The  diagram  is  combined  with  the  diagram  for  the 
horizontal  distance  worked  out  above  as  the  two 
quantities  H  and  V  are  always  computed  together. 
The  reader  will  readily  see  that  the  same  transforma- 
tion was  necessarily  applied  to  both  parts  of  the 
diagram  to  improve  the  arrangement  of  the  values  of 
the  vertical  angle.  Figure  50  shows  the  combined 
diagram. 

It  is  not  of  course  necessary  that  the  two  straight 
scales  be  parallel  when  there  is  but  one  curved  scale. 
Below  is  given  an  example  where  the  two  straight 
scales  are  graduated  on  the  axes  of  coordinates. 

Example  33. — The  formula  for  the  mean  hydrauhc 
radius  of  trapezoidal  sections  of  canals  may  be  written 
li(b  +  h  cot  4>) 
^  =  b  +  2//V1  +  cot-  <!> 

Where  R  =  mean  hydraulic  radius 
h  =  depth  of  water 
b  —  width  of  canal  bottom 
0  =  angle  which  the  side  slope  makes  with 
the  horizontal. 


Write  ^  = 
equation  is 


and 


li    and    a    final    determinant 


K 

1 

0 

1 

tan  </. 

1 

The  defining  equations  can  be  written  without  the 
use  of  scale  factors  as  follows 

a-  =  0  y  =  K 

X  =  l^i  —  stc  4>  y  =  —  tan  </> 

The  values  of  the  angle   <i>  are  graduated  on  an 
equilateral  hyperbola  crossing  the  X  axis  at  a;  =  —  H  ■ 
The  diagram  is  shown  in  Fig.  51. 
When  h  is  given  there  is  of  course  not  much  disadvan- 
tage in  computing  R  from  the  value  of  -r  • 

If  it  were  required  to  use  scale  factors  to  establish 
a  diagram  for  this  equation  above  which  has  actually 
the  form 

0         g,        1 

/o        0  1=0  (22) 

h  g3  1 

or  expanded,  the  form 

gl/3+/2g3-/2gl   =    0  (23) 

then    the    projective    transformations    developed    in 


treating  Equation  (18)  are  available.     The  equations 
were 


(m2  —  lil)x  -\-  Ml 


>'l 


(21) 


(mi  —  Ml)^+  Ml 
Then  the  new  defining  equations  corresponding  to 
Equation  (22)  would  be 


0 

(M2 


5M2/2 
-    Atl)/2  + 
5M2/3 


(M2 


l)/3  + 


MlM2g3 


0/3  +  Ml 


Another  simple  reduced  determinant  equation  for 
which  there  are  two  parallel  straight  scales  and  a 
curved  scale  is 

-1  «i         1     I 

1         g2        1     1=0  (24) 

I         /a  g3  1      I 

The  expanded  form  of  this  equation  is 

(gi  +  g2)  -  hig.  -  g.)  -  2g3  =  0  (25) 

The  defining  equations  for  the  curved  scale  will 
undergo  a  change  should  scale  factors  be  introduced 
in  the  equations  of  the  straight  scales.  Suppose  that 
it  is  desired  to  have  the  two  parallel  scales  at  equal 
distances  8  from  the  Y  axis  and  to  use  the  scale 
factors  Ml  and  m2  respectively.  The  first  defining 
equations  as  before  have  the  form 

X  =  -8  y  =  Migi 

X  =       8  y  =  H2g2 

and  the  third  defining  equations  must  be  assumed  to 

have  the  form 

X  =  F3  y  =  Gi 

where  Fz  and  d  are  to  be  functions  of  /s  and  gz  alone 
and  will  involve  the  constants  5,  mi  and  m2.  The 
reduced  determinant  form  of  the  equation  will  then 
become 

I      -5      •     Migi  1      ! 

I  5  M2g2  11    =    0 

\         Fz      Gz  1     I 

and  upon  expanding  there  results 

5(Migi  +  M2g2)  -  Fzijiigi  -  H2g'i)  -  26G3  =  0 
But  from  Equation  (25) 

g2   -    2gz  +  fzg2 

which  substituted  in  the  equation  above  yields 


g2   5m2+5m 


I+/3 


I+/3 

8lJ.lg3   —   Fzfllgz 


fz-l  ^=J    =  ^ 

and  this  equation  must  be  true  for  every  value  of  ga. 


56 


DESIGX  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


Diagram  for  the 
'ean  Ht/draulic  Radius  of 
Trapezoidal  Canals 


ALIGNMENT  DIAGRAMS  OR  COLLI  NEAR  NOMOGRAMS 


57 


Therefore    the   coefficient   of   gi   and    the    term  not 
involving  gi  must  vanish  identically,  that  is 


b 

(mi 

+  M2)/3  +  (mi  -  M2) 

(m: 

-    M2)/3  +    (mi   +   M2) 
2miM2^3 

(mi  —  M2)/3  +   (mi  +  M2) 

and  the  changed  form  of  the  equations  of  the  curved 
scale  of  the  diagram  become 

(ah  +  ix-2)fi  +   (mi  —  M2) 


M2)/; 
2MiM2g; 


+  (mi  +  M2) 


(26) 


(mi   —    M2)/3  +    iP\   +  M2) 

The  above  equations  are  very  important  for  the 
construction  of  diagrams  discussed  in  the  succeeding 
sections  of  this  book.  They  are  the  result  of  the  appli- 
cation of  the  projective  transformation 

^  (mi  +  tii)x  +  (mi  —  M2) 


M2)-V  + 

2miM2}' 


(m,  +  M2) 


(27) 


''  (mi  -  M2)X  +  (mi  +  M2) 

to  the  points  of  the  figure  as  originally  defined. 

It  is  seldom  that  the  functions  in  an  engineering 
formula  similar  to  Equation  (8)  are  of  such  general 
form  that  more  than  one  curved  scale  results  in  the 
diagram  and  indeed  no  rule  can  be  given  for  the 
introduction  of  scale  factors  when  the  defining  equa- 
tions are  of  the  most  general  form.  It  is  impossible 
for  example  to  introduce  different  pairs  of  values  of 
8  and  m  in  the  first  two  defining  equations  for  if  no 
restriction  were  placed  on  the  nature  of  the  functions 
/ii  /a,  ^1,  §2,  the  first  two  curves  originally  defined 
might  intersect  in  one  or  more  points  and  to  use  the 
scale  factors  5i  and  mi;  and  §2  and  m2  would  generally 
demand  that  the  same  points  of  intersection  of  the 
curves  supporting  the  two  original  scales  must  move 
in  different  directions  at  the  same  time  and  take  new 
positions.  It  is  necessary,  consequently,  to  leave  to 
the  reader  the  introduction  of  desirable  scale  factors 
in  those  cases  of  Equation  (8)  not  already  treated. 
It  will  be  necessary  to  take  advantage  of  the  particular 
form  of  the  individual  equation  in  hand  and  to  use  the 
general  methods  here  developed.  A  thorough  under- 
standing of  the  use  of  the  projective  transformation 
which  is  developed  in  Appendi.x  B  is  very  helpful. 

15.  Diagrams  of  Alignment  with  One  FixedPoint. — 
Equation  (16)  of  Article  13  may  be  written 

J\  -  M'2  =  0  (28) 

and  given  the  determinant  form 

/l  g'2  1        I 

/3        1         1=0  (29) 

0        0         1 


Then  the  three  pairs  of  equations 


=  1 

=  0 


define  respectively:  All  the  points  of  the  plane,  all 
the  points  of  the  line  y  =  1  (a  straight  function  scale), 
and  the  origin.  A  diagram  may  be  designed  on 
suitable  cross-section  paper  with  abscissas  as  values 
of  /i  and  ordinates  as  values  of  g^  and  inscribed  with 
corresponding  values  of  the  variables  Si  and  22,  and 
with  a  scale  of  the  function /s  on  the  line  y  =  1 .  Then 
the  values  of  2  which  constitute  a  solution  of  Equation 
(29)  are  collinear.  Since  the  index  always  passes 
through  the  origin  it  may  be  scratched  on  a  piece  of 
celluloid  pivoted  at  that  point. 

Example  34. — The  formula  of  Francis 

q  =  3.33S^^ 

yields  a  diagram  of  the  above  type  and  the  defining 
equations  are  conveniently: 


10 
3.335* 

0 


See  Fig.  52. 

Whenever  the  functions/i  and  g^  are  linear  functions 
of  the  variables  21  and  22  respectively  ordinary  cross- 
section  paper  may  be  used  quickly  to  establish  the 
desired  diagram.  It  is  of  course  optional  which 
function  /s  or  go  is  used  in  the  first  row  of  the 
determinant. 

By  using  logarithmic  cross-section  paper,  equations 
of  the  form 

/.  -  g^'  =  0 
may  readily  be  solved  for  a  limited  range  of  the  vari- 
ables   involved    to    almost    any    desired    degree    of 
accuracy.     Passing  to  logarithms 

log/,  -  23  logg2  =  0 

and  with  the  defining  equations  from  the  determinant 
of  Equation  (29)  there  results 


log/i 

23 

0 


y  =  log  g2 
y=  1 
v=  0 


There  is  an  ordinary  scale  on  the  line  y  =  1  and  the 
logarithmic  cross-section  paper  is  inscribed  with 
values  of  Z\  and  22. 

Example  35. — Frequently  in  thermodynamics  the 
equation 

PV"  =  C 


58 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


-     .  :,_   _r               1  ' 

_    n ^ 1  1  1  1  1  1   1   1  1   |g 

1 

1 

^ 

1  1 

T 

1 

— 

;               1   1                         :                     j 

' 

o 

1               !                     !   ' 

!      1      i         !'                       '                  1 

o'«-  .Ml              '                   '    ^          -L    . 

' 

"  ±   ' 

\ 

i                                                                                        o 

\                 1                 '                j 

1 

\                            i                               1 

'                                                         1 

^-                               : 

1  ~r                                               !£ 

i                         1                     M 

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■ 

T 

1         1  ! 

'    \       !   1      ' 

I          !      \     i    '          '                   ! 

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S 

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4-        -u  _L    -V   !  1            JL 

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1                       \: 

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1      ^                 =t"       -                  -        ' 

_ 

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i                             i     o 

X 

it                                                                               o 

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X: 

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X 

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a 

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t 

so       3^ 

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0-  — ±___::  :±:  :  ::    :" 

::  '::    ::::  :::x::  liSo 

ALIGNMENT  DIAGRAMS  OR  COLLINEAR  NOMOGRAMS 


59 


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///, 

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/// 

V/// 

i^ 

// 

/// 

//// 

c> 

M-^ 

/// 

vZZ 

■" 

o^Z- 

/  / 

//// 

''TuL 

-^^^'^ 

/  / 

/  '  /  / 

'/i 

-^s/. 

-// 

/  / ' 

// 

u^/ 

'// 

//// 

/ 

-  F-7- 

/  / 

fo/sZ^ 

/  / 

/// 

-  /y- 

//, 

// 

^/^iz 

//, 

V 

"-/6 

'/A 

/ 

-.^'6 

'// 

"?      ?7- 

/ 

;:/Z'^ 

JL 

•^  /v 

./'/ 

~ 

;# 

go 

OK 


>OC5C>    c>     <=> 

/o'oiodH  vi>    u-J     ^ 


>-|>^ 


60 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


arises,  and  with  a  given  set  of  values  of  C  and  w  it  is      The  defining  equations  of  a  suitable  diagram  of  the 
desired  to  find  an  indefinite  number  of  closely  deter-      type  under  discussion  are  then 
mined  points  on  the  curve  plotted  with  P  as  ordinates 
and  1'  as  abscissas.     It  is  usually  desired  to  find  addi- 
tional pairs  of  values  Pi  and  Vo  to  satisfy  the  equation 

PiFi"  =  PiVi"  and  the  diagram  is  shown  in  Fig.  53. 


=  log  AP 

y  =  log  AV 

=  —n 

y=  1 

=  0 

^  =  0 

S5      6.0    t.5    70  IS    8.   65  90  9.5    10 


.10      .11      .IZ    .13    .14-  .15  .16   .0  .16  .19 


where  Pi  and  Fi  have  been  determined.     This  equa- 
tion may  be  written 

[log  P,  -  log  P,]  +  «[log  Vi  -  log  V2]  =0 


logAP+ wlogAF=  0 


While  the  diagram  consists  essentially  of  the  loga- 
rithmic cross-section  paper  with  the  scale  for  n  and  the 
origin  clearly  marked  upon  it,  in  this  figure  the  various 
positions  of  the  index  have  been  drawn  in  as  straight 
lines  for  n.  This  is  allowable  since  the  equation 
log  AP-t-  «log  AF  =  0 


ALIGNMENT  DIAGRAMS  OR  COLLINEAR  NOMOGRAMS 


61 


10.0-1 

9.0- 

8.0- 

7.0- 

G.O- 

5.0- 

4.0- 
5.CM 


l_?.o- 


C5 

^1.0- 
^0.9- 

1  0.8- 
.E  0.7- 
1   0.6- 


n    0,5- 

o 


0.4- 


0.3- 


0.7- 


Diagram  for 
yL8G 

JUS 


H=  0.38 
and 


o 


20-1 
15.0- 

10.0- 

9.0 
8.0H 

TO 

G.O 

5.0- 

4.0 
3.0- 

7.0- 


1.0- 
0.9- 

0.8- 

0.1- 

0.&- 

0.5^ 
0.4- 

o.s- 


plO 
-9 
8 
1-7 


500-- 

400 

300 


tiooo 


700- 


4-500 
400 
300 

-1-200 
150 


50- 
40 -f 
30- 

70- 

^   10- 


I00-i|=:8O 
-60 

-40 
-30 

.70 


o 

1/5 

gO.5 

=E  0.4 

:^  0.3 

0.7 

O.I 


l?-[-G 

GG- 

Go4-5 

54 

48- 

42 

3^-3 
30- 

24- 
77- 
70- 

18- 

-f- 

■z    lP  <o  'G- 
^    ^-^15- 

t-7  :§  ^i4H 

J0-]  ETI2-I-I.0 

^  °-^  -5  *^  Q  J 

0.3-1^0.4  d-^ 
0.3 


-10 


o 

31 

^   5 

fc    4-^4 
^3 

S    2 


c 
o 

G. 


:-o.2 


04 
0.3 

0.2 -z 
0.1 


|8 
;^7 


-O.IO 

0.5 -^n%7 


..07 

0.3— -005 

-0.04 
-0.03 

■  0.02 


L 


0.01 


0.2- 


a. 

CI. 


0.9    I 
0.8  Q 


■07 
-O.G 


-0.5 


0.4 


■0.3 


0.2 


0.1 


DESIGN  OF  DIAGILAMS  FOR  ENGINEERING  FORMULAS 


is  also  in  the  special  form  of  equation  (5)  Article  8 
which  was  shown  to  yield  a  family  of  radial  straight 
lines  through  the  origin.  Since  the  range  of  numbers 
for  n  is  small,  the  useful  area  of  Fig.  53  is  rather 
limited.  The  two  squares  OABC  and  ODEF  nearest 
the  origin  have  been  superimposed  and  drawn  to  a 
larger  scale  in  Fig.  53a.  This  gives  a  more  convenient 
and  accurate  diagram  which  is  in  a  form  suitable  for 
rapid  plotting  of  the  PF"  =  C  curve.     The  inscribed 


values  of  ^   and  77- 
^2  Vi 


found  on  all  four  sides  of  the 


of  the  index  are  drawn  on  transparent  celluloid, 
together  with  the  scale  x  =  a,  y  =  1  and  the  resulting 
"index"  arranged  to  permit  translation  along  the  X 
axis  over  the  cross-section  sheet  defining  the  lines 
X  =  c,  y  =  b.  For  then  by  determining  a  first  prod- 
uct ci  =^  fli^i  on  the  X  axis  with  the  "index"  axes 
coincident  with  the  axes  of  coordinates  (on  the  cross- 
section  paper  beneath)  there  may  be  added  to  Ci 
the  product  Cz  =  0262  by  translating  the  "index" 
to  the  point  Ci  on  the  X  axis  and  then  reading  c  = 
Ci  +  C2  on  the  X  axis  at  the  foot  of  the  ordinate 


D/AGMMFOR  ^^l^^^^^ 
Redangular  Orifice  under  Low  Mead . 
V=  Average  VelocHy^  ft  per  sec. 


Values  of  Vjff.persec. 


logarithmic  cross-section  paper  of  Fig.  53a  apply  only 
to  the  half  of  the  diagram  between  them  and  the 
line  n  =  I. 

Equation  (29)  when/i  =  c,  g2  =  b,  and/3  =  a,  will 
determine  a  multiplication  diagram  for  the  product 
ab  =  c.  If  the  various  positions  of  the  index  are  drawn 
through  the  origin  for  products  of  integers  and  tenths, 
a  convenient  multiplication  diagram  results  for  small 
numbers.  By  suitable  choice  of  scale  factors,  dia- 
grams for  products  of  special  ranges  of  numbers  may 
be  prepared. 

A  useful  extension  of  this  diagram  results  if  the 
radial  Unes  through  the  origin  showing  the  positions 


through  the  intersection  of  the  "index"  line  a-^  "and 
the  second  multiplicand  line  corresponding  to  b^. 
(See  Problem  12,  Chapter  III.) 

Problem  1. — The  formula  of  Francis 
q  =  3.335^^ 
is  of  the  form  of  Equation  (12).     Construct  a  diagram  for 
this  formula  using  suitable  scale  factors.     Another  method 
of  treating  this  same  formula  is  given  in  Section  13  and 
still  another  in  Section  15. 

Problem  2. — The  formulas 


E  =  0.38  -^26  ^"d  Q 


dW 


ALIGNMENT  DIAGRAMS  OR  COLLI  NEAR  NOMOGRAMS 


100-1 


Dio^ram  fbr 

Mean  Temperafure  Difference 

Formula 


90-^ 


d  = 


T,-T2 


%  I 


^  80- 

o 


^10- 


g  GO' 


40^ 


30-^ 


?0- 


64 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


were  combined  in  Fig.  30  and  four  systems  of  lines  resulted. 
In  Fig.  54  are  shown  four  straight  line  scales  for  the  same 
equations.  Write  the  equations  necessary  to  describe 
the  method  of  constructing  Fig.  54. 

Problem  3. — Show  how  Francis'  formula  in  Example  27 
is  given  the  determinant  form  there  used  by  virtue  of 
Equations  (17)  and  write  three  other  possible  determinant 
forms. 

Problem  4. — Test  the  accuracy  of  Figs.  48  and  49  by 
setting  up  quadratic  and  cubic  equations  with  known  roots. 

Problem  5. — Show  how  the  determinant  form  of  Example 
30  was  obtained  by  the  laws  of  determinants.  (See 
Appendix  A.) 

Problem  6. — Draw  a  diagram  for  the  above  cubic  equa- 
tion based  on  a  reduced  determinant  form  of  the  equation 
analogous  to  that  used  for  the  quadratic  equation. 

Problem  7. — A  formula  for  the  approximate  area  of  a 
circular  segment  of  radius  R  and  with  height  H  is 


4^2    /2^ 


0.608 


It  is  possible  to  reduce  this  formula  to  the  form  of  Equation 
(18)  above.     Construct  a  diagram  for  it. 

Problem  8. — Develop  equations  for  introducing  scale 
factors  in  Equation  (22)  by  the  methods  used  in  the  other 
cases  of  the  present  chapter. 

Problem  9. — Write  the  set  of  equations  showing  the 
analysis  of  Figs.  55  and  56. 

Problem  10. — Show  that  the  formula  for  the  mean 
hydraulic  radius  of  trapezoidal  sections  of  Example  33 
may  be  represented  by  a  diagram  with  two  straight 
scales  and  a  third  scale  inscribed  upon  a  circle.  The 
necessary  reduced  determinant  form  may  be  derived 
from  the  determinant  equation  of  Example  33  by  the 
laws  of  determinants. 

Problem  11. — Show  that  the  mean  hydraulic  radius  of 
the  circular  segment  of  Fig.  41  has  the  form 
M.H.R.  =  HrfiK) 

where  f(K)  means  as  before  a  function  of  the  ratio  -^,  and 

design  a  suitable  diagram  for  the  mean  hydraulic  radius 
of  circular  sewers  flowing  at  any  depth. 


Problem  12. — Frequently    in    civil    engineering    "end 
areas"  are  determined  by  a  formula 
A  =  }4[(x2-x,)(y,  +  V,)  +  (.V3  -  x,)iy,  +  y,) 

+  .  .  .  +(A-,-.r„)(yi +  >>„)] 
when  Xi  represents  distances  from  a  center  line,  and  y,  cuts 
and  fills  at  the  corresponding  points;  construct  a  diagram 
with  a  sliding  index  to  compute  A  for  values  of  Xi  and  yi 
varying  by  tenths  up  to  20  feet. 

Problem  13. — Devise  a  combined  diagram  to  handle  the 
following  relation  in  thermodynamics 

Ti  ^  \pj    "    "  \YJ 

Problem  14. — Construct  a  diagram  of  three  parallel 
straight  scales  for  the  expression  of  Problem  14  of  Chapter 
II. 

Problem  15. — The  tractive  resistance  R  in  pounds  of  an 
automobile  of  weight  W  pounds  when  moving  at  a  speed  of 
V  miles  per  hour  is  given  by  Prof.  E.  H.  Lockwood  as 

i?  =  15  +  mbW  +  .075F2 
Construct  a  diagram  for  this  formula. 

Problem  16. —  Construct  a  diagram  consisting  of  four 
parallel  straight  line  scales  upon  which  the  collineation 
of  four  points  will  serve  to  solve  the  two  equations 

P.    =     </PW2 

Pt  =  \/p^' 

as  used  in  determining  the  ideal  intercooler  pressures  in  a 
three  stage  air  compressor.  Pi  =  initial  pressure,  Po  = 
final  pressure,  Pa  =  first  intercooler  pressure.  Pi,  = 
second  intercooler  pressure,  all  in  pounds  per  square  inch 
absolute. 

Problem  17. — Using  the  methods  of  Article  15  investigate 
the  necessary  scale  factors  to  establish  a  working  diagram 
for  finding  the  present  value  of  one  dollar  at  interest  rates 
of  2  to  8  per  cent  and  for  periods  of  5  to  20  years  by  the 
formula 

r"  =  (1  +  /)- 
where    V  =  present  value  of  1  due  in  n  periods  or  years 
i  =  interest  rate  (decimal) 
n  =  number  of  interest  periods  or  years. 


CHAPTER  rV 
ALIGNMENT  DIAGRAMS  FOR  FORMULAS  IN  MORE  THAN  THREE  VARIABLES 


16.  Binary  Function  Scales  and  Curve  Nets.— 

Suppose  that  in  the  XY  plane  there  are  plotted  two 
systems  of  curves, 

(t>iixy)  =  2i        <i>2ixy)  =  22  (N) 

as  shown  in  Fig.  57. 

Through  every  point  P  of  the  plane  will  pass  a  curve 
of  each  system  inscribed  with  its  corresponding  value 
of  z.  This  configuration  of  curves  will  be  called  the 
curve  net  Nu  for  Si  and  Zo.  A  line  perpendicular  to 
OX  is  seen  to  cut  out  an  indefinite  number  of  pairs 
of  values  of  Zi  and  Zi.  Every  point  M 
of  OX  may  thus  be  regarded  as  supplied 
with  all  the  pairs  of  values  of  Zi  and  Z2 
which  correspond  to  the  curves  intersect- 
ing on  the  line  PM.  These  value  pairs 
cannot  all  be  written  at  the  point  M  but 
are  nevertheless  definitely  attached  to  it. 
Furthermore,  given  the  value  of  Zi  there 
is  but  one'  corresponding  value  of  S2  to 
be  found  upon  PM. 

If  now  every  point  M  on  OX  is  regarded 
as  supplied  in  this  way  with  its  values  of 
Zi  and  zo,  the  line  OX  becomes  a  certain 
kind  of  scale.  Each  length  OM  deter- 
mines uniquely  a  line  MP  on  which  lies  a 
certain  set  of  values  Z1Z2. 

Let  OM  =  xi  and  consider  the  line 
Xi  =  Xi  and  the  curves 

4>iixy)  =  zi  <t>'i{xy)  =  Z2 

Eliminating  x  and  y  from  these  three 
equations  yields  an  equation  in  Zi,  02  and 
x  which  may  be  written 

/(Z,Z2)    =   X, 

All  the  values  of  Z1Z2  which  satisfy  the  above  equations 
belong  to  the  point  M. 

Conversely,  given  a  value  of  Zi  (or  Z2)  and  the  point 
M  (which  is  equivalent  to  assuming  the  value  of  Xi), 
there  is  in  general  but  one  value  of  Z2  (or  Zi)  which  will 
satisfy  the  last  equation.  It  is  thus  convenient  to 
define  the  configuration  of  Fig.  57  as  a  binary  function 

'  If  the  line  PM  intersects  the  curve  corresponding  to  Zi  in  n  points 
there  will  of  course  correspond  n  values  of  Zj,  etc. 


scale  for  the  function  fu  on  OX  which  is  called  the 
support. 

Similarly  eliminating  x  from  the  equations  (N) 
leads  to  the  result 

g(ziZ2)  =  y 
The  curve  net  of  Fig.  57  thus  completely  determines 
also  a  binary  function  scale  for  gio  on  OY. 

Frequently  a  pair  of  functions /12  and  gn  occur  in  an 
equation  of  four  variables  for  which  a  diagram  is  to 
be  constructed,  and  when  the  equation  is  put  into 


Fig.  57. 
the  determinant  form  analogous  to  Equation  (8)  it  is 
necessary  to  interpret  the  defining  equations 

X  =  /i2  y  =  gn 

It  is  evident  from  the  foregoing  that  these  two  equa- 
tions define  a  curve  net.  It  is  merely  necessary  to 
eliminate  Z2  and  Zi  successively  and  there  is  obtained 
again 

<t>iixy)  =  zi  <l>n{xy)  =  Z2 


65 


66 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


Since  the  only  necessary  equations  for  a  binary  scale 
on  the  A'  axis  are 

X  =  /i2  and  y  =  0 
they  are  called  the  defining  equations  for  the  binary 
scale.  In  constructing  a  curve  net  Nu  for  this  scale 
it  is  seen  that  either  of  the  functions  (fi  or  (pi  may  be 
arbitrarily  chosen.  When,  however,  one  function, 
say  01,  is  chosen  the  other  function  02  is  determined  by 
eliminating  Si  from  the  equations 

<pi{xy)  =  Zi     and    f{ziz-i)  =  x 
Obviously  <f>i  cannot  be  a  function  of  x  alone.     Exam- 


ples below  will  show,  however,  that  a  suitable  choice  of 
the  arbitrary  function  aids  in  the  solution. 

The  binary  scale  is  really  then  a  special  case  of  a 
curve  net  resulting  when  either  of  the  functions  fn 
or  gi2  defining  a  curve  net  reduces  to  zero  or  any 
constant.  When  either  Zi  or  Zi  only  is  absent  from/12 
or  from  gi2  one  set  of  curves  in  the  corresponding  net 
will  be  a  system  of  parallel  straight  lines.  (For 
another  special  case  see  Article  23  of  Chapter  VI.) 
When  a  binary  scale  has  been  established  on  either 
axis  or  upon  a  line  parallel  to  either  axis  it  is  obvious 
that  the  necessary  curve  net  may  be  moved  by  trans- 
lation parallel  to  the  other  axis  provided  that  the 
straight  line  support  remains  fixed.  Obviously  then 
a  new  origin  of  the  axes  of  coordinates  may  thus  be 
chosen  for  plotting  the  necessary  curve  net  and  this 


method  is  sometimes  of  much  advantage  in  improving 
the  plan  of  the  diagram. 

17.  Collinear  Diagrams  with  Two  Parallel  Scales 
and  One  Curve  Net. — Consider  now  the  equation  in 
four  variables  Zi(i  =  1,  2,  3,  4),  which  may  be  given 
the  determinant  form 

/i  ^1  1 

/2  g2  1  =0  (30) 

/34  ^34  1 

A  simple  case  arises  when  this  reduced  determinant 
equation  may  be  written  analogous  to  Equation  (18) 
of  Chapter  III: 

1  gi  1     I 

0  g,  1=0  (31) 

/34  gZi  1        I 

When  this  equation  is  expanded  there  results 

g2+/34(gl-§2)-g34    =    0  (32) 

Assume  temporarily  that  the  scale  factors  are 
unity  and  there  results  a  set  of  defining  equa- 
tions from  Equation  (31): 

X  =  1  y  =  gi 

X  =  0  y  =  gi 

X   =  fsi  y   =    g34 

and  the  last  two  equations  define  a  curve  net. 
It  is  thus  necessary  to  study  a  collinear  nomo- 
gram or  diagram  of  alignment  consisting  of  two 
parallel  straight  scales  and  a  set  of  points 
defined  by  an  inscribed  curve  net.  To  each  point 
of  the  curve  net  corresponds  a  pair  of  values  2122 
attached  to  the  two  curves  passing  through  that 
point.  The  equations  of  the  curve  net  are  readily 
written  by  eliminating  z^  and  23  successively  from 
the  last  two  equations  and  they  become 

<^3(^>')  =  23  <l>i{xy)  =  24 

The  resulting  configuration  is  shown  schematically 
in  Fig.  58. 

Given  three  values  of  Zj,  the  diagram  of  Fig.  58  con- 
stitutes a  complete  graphic  solution  for  the  unknown 
value  of  z.  Suppose  that  Z4  is  unknown:  The  line 
P1P2  cuts  then  the  curve  Z3  in  the  point  P  through 
which  passes  a  curve  marked  24.  The  proof  that  this 
value  of  24  is  the  value  sought  is  left  to  the  reader. 

For  certain  equations  in  four  variables  there  is  thus 
realized  an  important  type  of  colUnear  diagram.  To 
be  solvable  by  such  a  diagram  an  equation  must  be 
reducible  to  the  form  (31).  Obviously  the  parallel 
scales  may  be  placed  at  a  distance  5  and  the  scale 
factors  Ml  and  1x2  employed  if  the  equations  of  the 
curve  net  are  determined  from  the  third  pair  of 
defining  equations  as  modified  by  the  Equations  (21) 


ALIGNMENT  DIAGRAMS  FOR  FORMULAS  IN  MORE  THAN  THREE  VARIABLES        67 


of  Chapter  III. 
equations 


There  results  then  for  the  defining 


=  S 

=  0 


y  =  Migi 
y  =  fi2g2 


(33) 


^^2/3  4 


A^lM2g34 


M2/34    —    Ml(/34   —    1)    '  /J2/34   —    A'l(/34   —    1) 

The  choice  of  the  constants  S,  m  and  ^2  should  of 
course  be  made  not  only  with  the  first  two  scales  in 
view  but  also  with  the  resultant  changes  in  the  curve 
net  fully  in  mind.  No  plotting  should  be  undertaken 
until  a  thorough  study  of  the  equations  has  been 
made  in  order  to  obtain  the  desired  range  of  values 
of  the  variables  involved  and  at  the  same  time  to 
reduce  as  far  as  possible  the  required  computation 
for  plotting  the  curve  net. 

Example  36. — A  very  good  illustrative  example  is 
afforded  by  the  complete  cubic  equation 

s'  +  aiz^  +  a.z  +  as  =  0 
which  may  be  given  the  determinant  form 


2-  +  2 
Whence  if  6  =  10,  , 
X  =  10 


2-  +  2 


10^ 

;-  +  = 


=  '  +   (73 


are  the  defining  equations  for  the  diagram  which  is 
shown  in  Fig.  59.  In  plotting  the  curve  net  for  the 
variables  s  and  a^  the  2  lines  parallel  to  the  Y  axis 
are  plotted  first  and  then  it  is  observed  that  the 
successive  as  curves  determine  regular  scales  on  each 
2  line  with  a  new  scale  factor  for  each.  It  is  only 
necessary  to  plot  the  curves  for  the  values  of  a^  equal 
to  —10,  0,  and  10  successively  to  determine  com- 
pletely the  system  of  curves.     The  scale  factor  on 

each  z  line  is  seen  to  be    ,  •    In  the  diagram  the 

2"  -f-  2 

dotted  line  shows  the  position  of  a  straight  edge  set  to 

solve    the    equation    2'  +  42^  -  4s  +  0.5  =  0.     The 

straight  edge  is  set  from  ai  =  +4  to  ai  =  —4   and 

gives  the  value  of  s  =  0.69  at  its  intersection  with  the 

curve  03  =  0.5. 

Another  simple  case  of  Equation  (30)  which  results 

in  the  same  form  of  diagram  is 

-1  g:  1 

1        ^2         1       =0  (34) 

/34  ^34  1 

The  expanded  equation  has  the  form 

2g34  +  /34(gl   -   g2)    -    (g.   +  g2)    =    0  (35) 


and  the  defining  equations  with  the  scale  factors 
determined  by  the  aid  of  Equations  (25)  and  (26) 
for  the  analogous  case  of  three  variables,  are 


(mi    +   M2)/34  +    (mi 


Mlgl 
M2g2 


(36) 


M2)/34  +    (mi   +   M2) 

y 


2MlM2g34 


(mi   —    M2)/34   +    (mi    +   M2) 

The  presence  of  the  constants  5  and  m  in  the  third  pair 
of  equations  allows  control  to  some  extent  of  the  dis- 
position of  the  resulting  curve  net.  Whenever  the 
scales  for  the  first  two  variables  extend  in  opposite 
directions  in  the  diagram  it  is  desirable  to  apply  a 
transformation  as  in  Section  12  of  Chapter  III.  This 
is  done  in  the  following  illustrative  examples. 

From  the  last  pair  of  defining  equations  in  (33)  and 
(36)  it  is  seen  that  whenever  Z3  or  Z4  is  absent  from  734 
there  results  a  system  of  straight  lines  parallel  to  the 
Y  axis  and  they  are  determined  by  a  scale  on  the  X 
axis  most  conveniently.  Whenever  /34  or  ^34  is  zero 
(or  when  734  is  constant)  there  result  the  defining 
equations  of  a  binary  scale  on  the  Y  axis  or  on  the  X 
axis  (or  on  the  line  x  =  constant)  respectively. 
Another  special  case  occurs  which  leads  to  a  curved 
binary  scale  and  is  discussed  in  Chapter  VI. 

Example  37. — As  an  illustrative  example  of  Equa- 
tion (34)  consider  Kutter's  formula  for  the  flow  of 
water  in  open  channels, 

1.81132    ,  0.00281 


41.6603 


V  = 


1+    41.6603  + 


0.00281]    n 
S      \VR 


Vrs 


Where     V  =  velocity  in  feet  per  second 

S  =  tangent  of  inclination  of  surface 

R  =  mean  hydraulic  radius 

n  =  Kutter's  coefficient  of  channel  bottom. 

The  above  formula  may  be  modified  by  setting 

1,000 
outside  the  radical.'     There  results 
1.81132 


44.4703  + 


V  = 


1  + 


44.4703  w 


-VRS 


hich,  if  44.4703 

-1 

0 


VR 

a,  and  1.81132  =  b,  reduces  to 
V  0  I 


0 


-VS  1 

-niVR  +  an)       0  {an  +  b)R       \ 

This  substitution  is  known  as  Flynn's  modification  of  Kutter's 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


ALIGNMENT  DIAGRAMS  FOR  FORMULAS  IN  MORE  THAN  THREE  VARIABLES 


as  a  first  determinant  form.     The  reduced  determi 
nant  form  is  then  found  to  be 

-1  V  1 


[an  +  b)R 


1  -VS 

-n(VR+an)     ^ 


ian  +  b}R  +  n{VR+an) 
The  defining  equations  written  from  Equations  (36) 


above  are 
X  =  -S 
X  =       5 


2VS 


X  =      5- 


b)R  -  uiu(VR  +  an) 


i{an  +  b)R  +  fjLMVR  +  an)    ' 
For  convenience  then  in  plotting  there  may  be  chosen 

S  =  10,        Ml  =  0.8        M2  =  80.0 
whence  the  scale  equations: 

a;  =  -10  y  =     0.8F  _ 

a;  =       10  y  =  -gOVs 

(an  +  b)R  -  lOOniVR  +  an) 

{an  +  b)R  +  100n{VR  +  an)  ■'  " 

Since  ^34  is  here  zero  there  is  a  binary  scale  on  the 

X  axis.     One  system  of  curves  in  the  net  defining  the 

binary  scale  may  well  be  chosen  as  the  parallel  lines 

y  =  2VR 
and  there  follows  upon  eliminating  R  the  cubic  curves 
for  n 

_       (an  +  ^>)v^  -  20(yn(y  +  2an) 
""  ~  ^^{an  +  b)y'  +  200n(y  +  2an) 
All  these  cubics  pass  through  the  point  x  =  —  10, 
y  =  0  and  are  asymptotic  to  the  vertical  linex  =10. 
(See  Fig.  60.) 

The  V  and  5  scales  would  naturally  lie  in  opposite 
directions  from  the  A'  axis  but  to  secure  a  better 
disposition  of  these  scales  and  thus  reduce  the  size 
of  the  sheet,  they  have  been  moved  by  using  the 
projective  transformation 

Xi  =  X  yi  =  X  +  y  +  10 

which  moves  all  points  along  their  ordinates  a  distance 
equal  to  the  abscissa  plus  10.  Thus  the  line  y  =  0 
becomes  the  line  y  =  x  -\-  10  which  is  the  line  MN 
in  the  diagram.  From  the  nature  of  the  binary  scale, 
however,  there  is  no  need  of  transforming  the  curve 
net  for  the  variables  n  and  R  and  this  has  not  been 
done  in  the  figure.  The  points  on  the  binary  scale 
are  simply  transferred  by  the  parallel  vertical  straight 
lines  from  the  X  axis  to  the  diagonal  which  thus 
becomes  the  new  support. 

Example  38. — Another  example  of  Equation  (34)  is 


afforded  by  Bazin's  formula  for  the  flow  of  water  in 
open  channels  which  is 

87 


-Vrs 


0.552  + 


VR 


where  V,  R,  and  5  have  the  same  meaning  as  above 
and  m  is  Bazin's  coefficient  of  bottom  condition. 

The  first  determinant  form  of  the  equation  may  be 
written 

1  V  0 

0  -V5  1  =  0 

(0.552i?  +  m)      0  S7R 

and  the  reduced  form  of  the  equation  is  then 


1 

S7R  -  0.552^yR  -  m 


V 

-VS 
0 


87  R  +  0.552VR  +  m 
The  corresponding  scale  equations  are 
x=  -b 


M.F 

-M-2V5 


^Mi87jg  -  M-2  (0.552 Vjg  +  m) 
Mi87i?  +  M2(0.552\/^  +  m)  ^  "  ^ 

There  is  again  a  binary  scale  on  the  X  axis  which  is 
determined  by  setting 
S  =  10,  Ml  =  0.8    M2  =  80  as  above  and  also  y  =  2-\/R 


whence 


x=  10, 


87y=  -  200(0.552y  +  m) 


87y=  +  200(0.552y  +  m) 
The  m  curves  of  the  corresponding  net  are  six  cubics 
for  Bazin's  six  values  of  m.  These  cubics  have  a 
singular  point  at  x  =  — 10,  y  =  0  and  are  asymptotic 
to  the  line  x  =  10.  Figure  61  shows  the  finished 
diagram  originally  plotted  with  a  modulus  of  one 
inch.  The  5  scale  and  the  binary  scale  have  been 
transformed  as  in  the  preceding  example. 

Equation  (31)   is  no  simpler  than  the  essentially 
equivalent  form 

fa        gzi         1 

/i  1  1=0 

h         0  1 

which  has  the  expanded  form : 

/34+g34(/2-/l)-/2    =    0 

and  for  which  the  corresponding  diagram  will  consist 
of  two  horizontal  (instead  of  vertical)  parallel  scales 
and  a  curve  net.  To  introduce  scale  factors  into 
the  corresponding  defining  equations  there  are  avail- 
able Equations  (21)  of  Chapter  III.  With  the  obvious 
changes  in  the  role  of  the  respective  coordinates  x  and 
y  there  results  from  these  equations 

IJ-\IJ-iX  5/J2V 


A'l)^ 


(m2 


i)y  + 


70 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


ALIGNMENT  DIAGRAMS  FOR  FORMULAS  IN  MORE  THAN  THREE  VARIABLES         71 


Fig.  61. — Diagram  for 


72 


DESIGX  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


Example  39. — One  form  of  the  fundamental  formula 
for  bond  calculations  is 
A 
C 


the  quantity 


tV 


-K4^) 


where  A  --= 
g  = 

C  = 

«  = 

purchase  price 

nominal  interest  rate 

effective  or  yield  rate  of  interest 

redemption  price 

term  of  bond  in  years 

•^  -  1  +  i 

This  bond  formula  has  the  reduced  determinant  form 

R                 0                  1 

g'                 1                  1 

=  0 

1-^;;]       1-^^ 

in  which  for  convenience  the  symbol  A~\  is  used  for 

\  —  V"  ,         S  —A 

— -. —  and  in  which  g   =    ^   and    R  =  j;-     This  is 

an  excellent  example  of  the  form  of  Equation  (31) 
above.  By  letting  /i2  =  1  and  tx\  =  ft.  and  substituting 
successively  in  the  modified  Equations  (21)  above  the 
respective  pairs  of  elements  from  the  determinant 
there  result  the  defining  equations 

x  =  R  y  =  0 

X  =  ixg'  y  =  5 

ixv"  8  A  ^1 

X  = 7- 1  y  = 7- 

1^   -  An  -'  1^   -  A-i 

The  curve  net  for  i  and  n  defined  by  the  last  pair  of 
equations  may  best  be  plotted  as  follows: 


The  ratio 


from  which 


Sx 


d.i-v" 


5x  —  ifiy 

and  when  this  value  of  v"  is  substituted  in  the  second 
equation 

bA-,  -5(1-1'") 


//i  -  1  +  v" 


there  results 

hix  -  1) 

which  defines  a  pencil  of  lines  through  x  =  1,  y  =  0 
as  the  i-lines.  The  equation  of  the  w-curves  may  be 
shown  to  be 

'  y 


/x  +  2y  -  1Y_   1- 

\        y        )        X 


if  both  n  and  8  are  taken  unity  but  it  is  not  necessary 
to  attempt  to  plot  from  this  equation.  Instead 
resume  the  equation 

X  /./f" 


is  tabulated  in  standard  works 
and 


on  bonds,  life  insurance,  etc.  and  is  designated  5 
rewritten 

5-  =  (1  +  0"  -  1 

"'  i 

From  these  double  entry  tables  when  /  is  constant  the 
values  of  Sn[  vary  for  n  only,  thus  the  equation 

X  M       ' 

will  determine  a  pencil  of  lines  through  the  origin 
varying  for  values  of  n.  These  lines  intersect  the 
corresponding  j-line  in  points  necessarily  on  the 
respective  w-curves.  Thus  the  n-curves  may  easily 
be  plotted. 

The    useful    range    of    values   of    the    ratio    ^    is 

from  say  "^^oo  to  ^^^loo  and  to  be  effective  in 
actual  bond  calculations  this  ratio  must  be  readable 
to  the  nearest  thousandth  or  tenth  of  a  per  cent,  conse- 
quently if  the  scale  should  show  one  per  cent  as  one- 
half  inch  the  effective  portion  would  be  22  ^  inches 
long  and  unity  would  be  represented  by  50  inches. 
The  choice  of  a»  and  5  must  then  be  made  and  it  is 
obvious  that  if  d  is  greater  than  unity  the  line  y  =  S 
on  which  is  to  be  shown  the  g'  scale  will  be  not  only 
off  any  drawing  of  dimension  less  than  50  inches  verti- 
cally but  also  (since  g'  will  never  be  much  greater  than 
Ho)  unless  n  is  large  g'  will  be  too  close  to  the  Y  axis 
to  appear  on  any  reasonably  sized  drawing  where 
unity  is  50  inches.  The  remedy  for  both  these 
troubles  is  to  choose  oblique  axes  with  a  very  acute 
angle.  When  this  is  done  and  with  8  =  %o  and  n  =2 
there  results  the  completed  drawing  shown  in  Fig.  62. 
It  is  observed  that  the  nominal  interest  scale  is 
inscribed  g  and  not  g'.  This  is  because  the  normal  case 
is  redemption  at  par  and  then  g'  reduces  to  g  the  nor- 
mal rate.  With  ;tt  =  2,  one  per  cent  on  the  g  scale  is 
represented  by  one  inch.  The  auxiliary  net  of  lines 
for  the  segregation  from  the  binary  scale  of  the  ratio 

pr  into    the  purchase  price  A    and   the  redemption 

price  C  is  effected  by  the  equations 
A 

The  choice  of  m  =  2  is  dictated  by  the  behavior  of 
the  n  curves  for  a  reasonable  range  of  useful  terms 
n  and  was  determined  only  after  several  trials. 

The  examples  here  worked  out  are  special  cases  of 
Equations  (31)  and  (34)  which  are  both  special  cases 
of  the  more  general  Equation  (30)  which  equation 
would  in  general  require  two  curved  scales  and  a  curve 


mA 


m  =  constant 


ALIGNMENT  DIAGRAMS  FOR  FORMULAS  IN  MORE  THAN  THREE  VARIABLES         73 


LEGEND 
A  =  Purchase  Price  or  Present  Value. 
C  =  Redemption  Price. 

§  =  Ratio  of  Ditiidend  to  Redemption. or  Diiiidend  Rate  with  Redemption  at  Par. 
i  =  Effective  Interest  Rate. 
n  •  Term  of  Years. 


Purchase  Price  A 
ift          o         >5         o 


DIAGRAM 

FOR  THE 

FUNDAMENTAL  BOND  FORMULA 

1 

c "    (i+ir  i 

FOR  AiNY  VARIABLE  IN  THE  FORMULA 


74 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


net  for  its  diagram.  Examples  encountered  in  prac- 
tice seldom  require  such  a  type  of  diagram  but  treat- 
ment of  the  scales  by  some  projective  transformation 
would  doubtless  be  needed  for  any  such  example. 

It  is  to  be  observed  that  whenever  one  set  of  curves 
in  a  curve  net  becomes  a  system  of  straight  lines,  then 
the  plotting  of  the  second  set  of  curves  can  often  be 
simplified  by  finding  indirectly  their  intersections  with 
this  plotted  line  system.  Such  was  essentially  the 
method  used  in  Examples  36  and  39. 

More  generally,  when  one  set  of  curves  of  a  curve 
net  has  been  plotted,  the  second  set  can  be  plotted  by 
determining  indirectly  points  of  intersection  with 
individual  curves  of  the  plotted  first  set.  To  do  this 
hold  constant  in  either  of  the  defining  equations  the 
value  of  the  variable  parameter  corresponding  to 
given  curve,  while  the  second  variable  corresponding 
to  the  desired  set  is  allowed  to  vary  and  draw  the 
resulting  lines  parallel  to  one  of  the  axes.  There  will 
thus  be  determined  on  the  plotted  curve  a  series  of 
points  of  intersection  corresponding  to  successive 
values  of  the  second  variable.  These  points  for 
constant  values  of  the  second  variable  on  successive 
curves  will  lie  on  a  curve  of  the  second  set.  In 
particular  if,  as  above  explained,  the  first  set  of 
curves  is  a  system  of  straight  lines,  then  the  curves 
of  the  second  system  can  always  be  found  by  plotting 
corresponding  points  of  intersection  of  this  first 
system  of  lines  with  the  system  of  lines  parallel  to 
either  one  of  the  axes.  In  Example  39  an  auxiliary 
set  of  lines  through  the  origin  was  used  to  advantage 
instead  of  the  parallel  lines  determined  by  a  defining 
equation. 

18.  Collinear  Diagrams  with  Three  Curve  Nets.— 
These  diagrams  and  indeed  diagrams  with  two  curve 
nets  are  largely  of  theoretic  interest  but  there  are 
special  cases  of  practical  value. 

Consider  first  an  equation  of  six  variables  in  the 
determinant  form 


M 

gl2 

1 

/34 

^34 

1 

Aa 

g66 

1 

(37) 


By  setting 


/./ 


1,3,5 
2,4,6 


there  are  obtained  three  curve  nets  constituting  a 
collinear  diagram  for  this  equation.  The  key  to 
the  solution  of  the  diagram  is  obvious  from  the 
schematic  Fig.  63.  Should  the  active  range  of  the 
variables  involved  determine  curve  nets  which  unduly 
overlap  or  confuse  the  diagram,  some  device  such  as 
different  colors  will  be  needed  to  make  the  drawing 


of  practical  value.  In  most  cases  that  occur  in 
practice  the  curve  nets  reduce  to  binary  scales  and 
seldom  are  there  more  than  two. 

Example  40. — As  an  illustrative  example  consider 
the  equation  for  the  angular  distance  2  of  a  celestial 
body  east  or  west  of  the  meridian  from  the  north 
point. 

where  L  =  the  latitude  of  observer 

p  =  the  polar  distance  of  the  object 
//  =  the  altitude  of  the  object 

5  =  H(h  +L  +  p) 

This  equation  may  be  solved  by  a  diagram  with  two 
binary  scales,  but  since  z  must  usually  be  determined 


cos 


/cos  5  cos  (S  —  p) 
cos  L  cos  // 


Fiu.  63. 

at  least  to  the  nearest  30  seconds  no  diagram  of  any 
practical  value  can  be  drawn  small  enough  to  repro- 
duce here  successfully.  The  variables  are  z,  S, 
(S  —  p),  L,  and  h.  After  squaring  both  sides  of  the 
equation  it  may  be  written  in  the  reduced  determinant 
form 

0  cos  5  cos  {S  -  p)  ] 


0 


=  0 


cos  L  cos  // 
1  -|-  cos  L  cos  h 

There  may  be  written  in  a  manner  analogous  to  Equa- 
tion (14)  of  Chapter  III,  the  defining  equations 
.V-  =  0  y  =  Ml  cos  5  cos  (5  -  p} 


cos  L  cos  // 


fin  cos  L  cos  //  +  n-i    ' 
The  first  two  equations  define  a  binary  scale  on  the  Y 
axis  and  the  variables  5  and  (S  —  p)  may  be  separated 
with  the  curve  net 

X  =  cos  S        y  =  fiix  cos  (5  —  p) 
which  gives  two  systems  of  straight  lines.     The  second 
equation  pair  defines  the  scale  of  length  M2  measured 
downward  on   the  line  x  =  b.     The  third  equation 


ALIGNMENT  DIAGRAMS  FOR  FORMULAS  IN  MORE  THAN  THREE  VARIABLES         75 


pair  determines  a  binary  scale  on  the  X  axis  and  is 
constructed  with  the  simple  curve  net 

—  All  cos  Ly 


Ly 


■  cos  h 


The  L  curves  are  then  equilateral  hyperbolas  passing 
through  the  origin  and  with  asymptotes  parallel  to 
the  coordinate  axes. 

Problem  1. — Consider  the  cubic  equation 

s^  +  a,c2  +  <j„c  +  03  =  0 
from  the  point  of  view  of  Equation  (34)  and  write  the 
defining   equations  of  a  diagram  with  parallel  scales  for 
02  and  as. 

Problem  2. — Consider  the  above  cubic  from  the  point  of 
view  of  Equation  (31)  and  write  the  equations  of  the  curve 
net  resulting  if  the  parallel  scales  are  for  the  variables 
02  and  as- 

Problem  3. — Assuming  that  the  first  two  pairs  of  defining 
equations  of  Equation  (34)  are  written 


X  =  —  5i                                  y  = 

^  Migl 

X  =        5o                                   y  = 

M2g2 

show  that  the  third  pair  of  defining  equations  are 

J  (52M1  +   5lM2)/34  +    (SoMl    - 

51M2) 

(/il    —   M2)/34  +  (mi   +  M2 

) 

2aIiM2J?34 

(mi  —  A'2)/3i  +  (mi  +M2) 

Problem  4. — The  area  of  a   trapezoidal 

section  of  an 

irrigation  canal  is  given  by  the  formula 

A  =  hb  +  h^  cot  0 

where  the  symbols  are  used  in  the  same  sense  as  in  Example 
33,  Chapter  III.  Construct  a  diagram  for  the  formula 
which  shall  have  two  straight  parallel  scales  for  A  and  b 
and  a  curve  net  for  h  and  <j>  consisting  respectively  of  lines 
parallel  to  the  Y  axis  and  hyperbolas  passing  through  the 
origin  and  tangent  to  the  X  axis  at  that  point. 

Problem  5. — Professor  C.  H.  Forsyth  has  given'  a 
formula  for  the  premium  or  discount  per  unit  on  a  bond 
if  the  "amortization  factor"  accumulates  at  a  rate  of 
interest  r  which  is  different  from  the  effective  or  yield 
rate  of  the  bond  i.  If  k  denotes  this  premium  or  discount 
then  with  the  notation  of  Example  39  and  redemption  at 
face  value  or  par  the  formula  is 

9  -  i 

where  the  changed  symbol  —  denotes  that  this  tabulated 
^Bulletin,  km.  Math.  Soc,  vol.  XXVII,  p.  451. 


quantity  ^  is  to  be  here  taken  at  the  rate  r.     Show  that 

this  formula  is  a  special  case  of  Equation  (37)  with  five 
variables  and  with  a  corresponding  diagram  which  consists 
of  a  straight  line  (cross-section)  net  for  g  and  k,  an  ordinary 
scale  for  ;  on  the  Y  axis  and  a  binary  scale  on  the  line 

-T  =  —  1  for  —  and  that  the  segregation  of  the  n  and  r 

Hues  in  the  binary  scale  net  can  be  obtained  by  setting 
X  =  r  —  \  and  plotting  the  n  curves  by  determining 
points  on  the  r  line  corresponding  to  changes  in  n  for 
constant  r. 

Problem  6. — In  the  above  problem  show  that  the 
equations  for  the  curve  net  for  g  and  k  can  also  have  the 
equations 


-bu^k 


(*12   -   Ml)^ 


y  = 


(M2 


.)fe+A 


if  scale  factors  6,  y.i,  ^2  are  introduced  by  Equation  (21)  of 
Chapter  III  and  that  consequently  an  ordinary  cross- 
section  net  for  g  and  k  results  when  jui  =  ix^. 

Problem  7. — The  so-called  premium  formula  for  bond 
valuation  is  with  the  usual  notation 

k=(g-  i)  /l'„-i 

where  A  %  indicates  that  ^  ^  is  to  be  evaluated  at  the  rate  i. 
When  the  bond  is  bought  at  a  discount  k  is  negative. 
Compare  this  equation  with  that  of  Example  39  and 
discuss  the  advantages,  Lf  any,  for  design  of  the  correspond- 
ing diagram. 

Problem  8. — Show  that  the  equation  for  the  «-curves 
in  the  curve  net  of  the  diagram  of  the  above  equation  are 
_i+_y_   . 
y     /        I  -\-  y  -  X 


given  by  the  equation 


(^)" 


Problem  9. — If  all  interests"  are  payable  m  times  a  year 
and  the  amortization  factor  accumulates  at  a  nominal  rate 
r  then  the  premium  formula  of  Professor  Forsyth  becomes 


where  5;;^  is  to  be  evaluated  for  mn  periods  at  rate  — 

and  where  ;  is  the  nominal  rate  to  be  realized.  Show  how 
this  equation  in  the  five  variables  k,  g,  j,  r,  n,  can  also  be 
diagrammatically  represented  for  values  of  m  from  w  =  1 
to  m  =  4. 

Problem  10. — Show  in  Example  36  how  the  cubic  curves 
for  as  could  have  been  plotted  by  first  plotting  a  system  of 
lines  parallel  to  the  X  axis  which  would  intersect  a  given 
2-line  parallel  to  the  Y  axis  in  points  corresponding  to 
values  of  03. 


CIL\PTER  V 
DIAGRAMS  OF  ALIGNMENT  WITH  TWO  OR  MORE  INDICES 


19.  Diagrams  of  Double  Alignment. — Sometimes 
a  formula  or  an  equation  may  be  given  the  form 

/l2  =  /34  (38) 

and  may  be  replaced  by  a  pair  of  equivalent  equations 

/l2  =    /2  34  =    fi 

where  h  is  an  auxiliary  variable.  Assume  now  that 
each  of  these  equations  can  be  represented  by  a 
diagram  of  alignment.  By  determining  the  value  of  h 
from  one  diagram  the  value  of  either  one  of  the 
remaining  variables,  say  Zz  or  Z4,  could  be  found  from 
the  second  diagram.  If,  however,  both  equations  can 
be  represented  with  the  same  scale  for  h,  a  single 


Fig.  63a. 

figure  with  four  z  scales  and  one  h  scale  would  consti- 
tute a  complete  diagram  for  the  original  equation. 

Such  a  diagram  is  called  a  diagram  of  double  align- 
ment or  a  diagram  of  double  collinealion.  The  scale  for 
//  is  called  the  hinge  or  pivot  scale  and  need  not  be 
graduated  unless  this  is  desirable  for  convenience  in 
locating  the  temporary  point  about  which  the  index 


4  sin  <t>       }i  cos  ^ 
A-  1 


is  turned  for  its  second  position.  The  type  of  diagram 
and  the  way  to  the  solution  is  shown  in  the  schematic 
figure.  Fig.  63a. 

The  diagram  is  often  more  conveniently  arranged 
when  the  part  including  the  Zi  and  z^  scales  is  super- 
imposed in  the  other  part  of  the  figure,  but  in  many 
cases  where  the  scales  are  on  parallel  supports  greater 


accuracy  and  ease  of  use  will  result  when  the  pivot 
scale  is  chosen  between  the  scales  for  each  part  of  the 
diagram  and  the  indices  are  placed  in  the  form  of  a 
letter  X  as  shown  in  Fig.  69,  page  8. 

Some  thought  should  be  given  to  the  way  in  which 
the  variables  are  grouped  on  either  side  of  the  equality 
sign  in  Equation  (38)  so  that  those  variables  which 
are  perhaps  more  closely  related  or  those  which  have 
about  the  same  range  of  numbers  may  be  used  in  the 
same  half  of  the  diagram.  It  is  usually  necessary  to 
use  different  schemes  of  scale  factors  for  each  auxiliary 
equation  and  the  only  restriction  on  the  equations 
is  that  they  be  of  type  (8)  of  Chapter  III.  It  is  always 
required,  however,  that  the  hinge  scale  have  the  same 
defining  equations  in  every  respect  in  order  that  the 
same  value  of  h  shall  be  determined  in  both  diagrams 
by  corresponding  values  of  the  variables. 

Example  41. — The  formula  of  Example  33  for  the 
mean  hydraulic  radius  of  trapezoidal  sections  of 
canals  will  be  arranged  as  an  example  of  the  diagram 
of  double  coUineation.     The  formula  is 


1  +  K  cot  <j> 


1  +2K  cosec  4>  b 

where  R  is  the  mean  hydraulic  radius,  H  the  depth  of 
water,  b  the  breadth  of  canal  bottom,  and  (t>  the  angle 
of  slope  with  the  horizontal.  The  formula  should 
first  be  r&written  in  the  form  of  Equation  38. 

R  _  J  _  sin  <j>  -f  Jv"  cos  0 
H~  ^  ~      sm4>  +  2K 

and  then  in  the  reduced  determinant  equation  forms 
0  h  1 


=  0  = 


1        -R 

ih  ' 

Since  K  will  never  be  greater  than  unity  it  is  evident 
that  the  unit  of  the  drawing  for  the  first  determinant 
must  be  fairly  large  and  it  may  be  chosen  as  4 
inches.  The  scale  for  the  angle  <t>  will  then  be  a  circle 
with  a  radius  of  one-half  unit  or  two  inches,  which 
is  ample,  as  the  angles  need  not  be  measured 
closer   than    degrees.     No   scale  factors  are  needed. 


DIAGRAMS  OF  ALIGNMENT  WITH  TWO  OR  MORE  INDICES 


77 


In  the  second  equation  it  is  seen  that  the  R  scale  may 
be  inconveniently  long  with  the  unit  of  the  drawing 
as  4  inches  and  yet  since  the  value  of  the  H  function 
is  never  greater  than  unity  the  horizontal  scale  must 
not  be  contracted.  What  is  needed  then  is  to  extend 
the  horizontal  scale  and  contract  the  vertical  scale 
and  at  the  same  time  leave  the  //  scale  unchanged. 


choosing  5  =  2.5. 
defining  equations 

rt  =  2.5 

x=      0 
5 


There  results  for  the  second  set  of 


H 

1.0  it 


0.5- 


Diagram  for 

R.-H'-ihtl- 


Jcosec^ 


The  result  is  accomplished  with  a  little  study  by  first 
writing  the  second  equation  in  the  form 


1 

0 

1_ 

l  +  H 


This  equation  is  of  the  type  (17)  of  Chapter  III  and 
Equations  (21)  are  applicable  to  the  defining  equations. 
Sufficient  contraction  \^^ll  result  if  ni  is  taken  }4,  then 
M2  must  be  unity  in  order  to  preserve  the  h  scale 
intact.  The  scale  for  H  may  be  extended  and  at  the 
same  time  the  position  of  the  R  scale  improved  by 


The  completed  diagram  is  shown  in  Fig.  64  which  has 
been  rotated  90°  to  improve  its  position  on  the  sheet. 
The  limiting  values  of  the  variables  chosen  here  are 
general  and  the  graduations  could  be  greatly  refined 
for  special  work,  for  small  drainage  ditches  for  example. 
Example  42. — Unwin's  formula  for  the  flow  of 
steam  in  pipes 


II'  =  87.5 


■I- 


Ppd" 


11  + 


3.6 


where  W  =  number  of  pounds  flowing  per  minute 
D  =  density  in  pounds  per  cubic  foot 
p  =  loss  of  pressure  due  to  friction,  in  pounds 
per  square  inch 


78 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


d  =  nominal  inside  diameter  of  pipe  in  inches 
L  =  length  of  pipe  in  feet, 
may  be  written 
11' 


8'Wf 


=    // 


VD 


^ 


3.6 


If  L  is  taken  as  100  feet  and  p  expressed  as  the  loss  in 
pressure  per  100  feet  of  length,  the  formula  is  similar 
to  Equation  (38) 
Tl' 


^  =  /,  =  ^/D 


S.75VP 
or  log  ir  —  log  8.75 


3.6 


\/l+    _ 
■4  log  p  =  \ogh 


MlogD  +  Mlog    j^3J5 


log  // 


Both  of  these  auxiliary  equations  are  similar  in  form 
to  Equation  (10)  and  yield  three  parallel  straight  scales. 
The  diagram  is  shown  in  Fig.  65  with  the  defining 
equations 

=  PI  log  ir 

=  -M2[log  8.75 +  H  log/'] 


X  =  -8i 

X  =         bi 

x=      0 


N.+^^ 


64 


The    proportion 


«i 


was      used 


log  \'D 

^  =  —  =l  =  ^=ii2 

&2  Hi  4  54  m 
to  give  a  sjonmetrical  diagram  with  convenient  scale 
lengths.  Since  D,  the  density,  depends  upon  the 
steam  pressure  the  corresponding  pressures  were 
plotted  in  place  of  the  various  densities.  To  the  scale 
for  W  was  added  a  scale  for  the  approximate  boiler 
horsepower.  The  indices  show  the  setting  to  deter- 
mine the  flow  in  a  6-inch  pipe  carrying  steam  at  140 
pounds  gauge  pressure  allowing  a  drop  of  3  pounds 
for  each  100  feet. 

Any  equation  or  formula  of  the  form 

(m,  »,  r,  s,  =  constants) 
may  be  replaced  by  the  equivalent  system  of  equations 

p'  6" 

and  two  corresponding  first  determinant  equations  are 

1  //  0 

0        -a""  1 

/>'  0  1 


1  h  0 

0        -q'  1 

b-  0  1 


(40) 


The  choice  of  the  reduced  determinant  forms  of  these 
equations  may  be  made  by  first  adding  either  the  first 
or  the  second  columns  to  the  third  to  form  a  new  third 
column  in  each  determinant.     The  choice  will  of  course 
be  made  with  a  view  to  the  economy  of  calculation 
for  the  resulting  scales  on  the  A'  or  Y  axis.     Equa- 
tions (40)  are  of  the  type  (14)  of  Chapter  III. 
The  above  equation  (39)  may  be  written 
m  log  a  —  r  log  p  =  log  h  =  s  log  q  —  n  log  b 
and  the  two  auxiliary  equations  will  be  similar  to  Equa- 
tion (10)  and  require  simply  four  logarithmic  scales 
on  as  many  parallel  straight  lines.     The  determinant 
equations  are 

- 1         m  log  a  1 

1     -rlogp  1=0 

0  logh  1     I 

1—1         s  log  q  1     ! 

1  -n]ogb  1      {  =  0 
1         0         logh  1 

and  the  defining  equations  of  Section  12  including  8 
and  M  apply  to  each. 

It  is  to  be  observed  that  no  plotting  on  the  //  scale 
is  necessary  but  the  same  value  of  h  must  determine 
the  same  point  on  the  h  scale  so  that  the  reduced  deter- 
minant forms  of  the  equations  must  both  result  in  the 
same  defining  equations  for  the  //  scale  even  though 
that  scale  is  not  graduated.  It  must  therefore  be 
borne  in  mind  that  the  choice  of  the  scale  factor  for 
the  h  scale  must  be  the  same  for  both  equations. 

Example  43. — As  an  illustrative  example  of  the 
above  Equation  (39)  Chezy's  formula  for  the  flow  of 
water  in  open  channels  may  be  studied.  The  formula 
is 

V  =  fv'^ 
where  V,  R,  and  S  have  the  designations  of  Example 
37  and  c  is  Chezy's  coefiicient.     The  form  for  the 
reduced  determinant  equations  may  be  taken 
1  h  1 

0 


0 

-V 

c 
c  +  1 

0 

1 

h 

0 

_rH 

5-w 

1+s-^ 

0 

It  will  be  necessary  to  graduate  the  scales  for  R  and 
V  and  for  S  and  c  on  the  same  axes,  and  this  will 
always  be  necessary  for  equations  of  the  tj-pe  (40). 
Since  the  c  and  S  scales  will  not  extend  beyond  unit 
distance  from  the  origin  it  will  be  well  to  make  6  as 


DIAGRAMS  OF  ALIGNMENT  WITH  TWO  OR  MORE  INDICES 


79 


-0.1 


20,000 


Diagram  for  -the 
FLOW  OF  STEAM  IN  PIPES 
Un win's  Formula 


w-87.sy£EiL 


W=  Flow  -  lb. per  mjn. 

D  =  Densi'iy  -  lb. per  cu.  FF. 

p  -  Pressure  Drop  -  lb. per  sq.  in. 

d  -  Inside  Diamefer  Pipe  -  in. 

L^LengihoFPipe-FI: 


-250 

-2Z5 

-200 

-180 

^ 

-m 

-- 

-140 

-120 

-100 

-90 

D 

-80 

O 

-TO 

.C 

-60 

d- 

Kf) 

-50 

ty 

-40 

-30 

oT" 

u 

3 

-20 

1?, 

-15 

d: 

-10 

E 

-^ 

-5 

^ 

To 

-2 

-0 

■£ 

IG.GOO-- 6,000 


12,000 

8,000 
6,000 


G.OOO 
4.000 
3,000 

4,ooa|- 1,000 

3.,000-- 1,500 
2,000- -1,000 


I.GOO- 

1,200- 

1,000- 

800 

GOO- 


GO 


c^30 


o   "20 


^    1.2 

0.8 
0.& 


10,000 


800 
-MO 
50(K^ 
400   ^ 

300   ^ 


50O::2'5a' 
^4^-200 
300- -150 

200-1-100 
IGO- 

% 
100 

80-H-o 


30 

-1-10 
15 


"18 --9      ^ 


4-2 
3-1-1.5 

1.0 
l.G-fo.8 


0.& 

04 
0.3 
0.4-L0.2 

Fig.  65. 


-12 
■10 

-8 
-T 
-& 
-5 

~^4 
-5 

u 

2     -E^ 

\%  J 
I 


-0.15 
-0.2 

-0.S 

-0.4 
-0.5 
-O.G 
-0.7 
-0.8 
-0.9 
-1.0 

-1.2 
-1.4 

-1.8 

-2.0 

-2.25 

-2.5 

-2.15 

-3.0 

■3.5 

■4.0 

-4.5 

-5.0 

1-6.0 
-7.0 
-8.0 
-9.0 
-10.0 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


large  as  the  sheet  will  permit.     Also  since  the  //  scale 
and  the  V  and  R  scales  are  in  opposite  directions  the 
X  axis  may  well  be  chosen  at  an  acute  angle  with  the 
Y  axis  at  the  outset. 
If  the  formula  is  written 


log  F  -  log  c 


log  R  +  M  log  5 


the  two  determinant  equations  are 
■  1  log  F         1 

1         -logc  1 

0  log  It  1 


-1         ^2\ogR         1 

1         i2log5  1       =0 

0  log  h  1 

and  the  general  arrangement  of  the  figure  would  be 
similar  to  Fig.  65  of  E.xample  42.  This  latter  form 
of  similar  formulas  usually  results  in  simpler  and  easier 
plotting. 

Some  formulas  in  four  or  more  variables  can  best  be 
handled  by  combinations  of  simple  diagrams  composed 
of  straight  line  systems  as  described  in  Chapter  II. 
The  use  of  such  diagrams  is  very  common  and  obvi- 
ously all  that  has  been  said  regarding  the  use  of  a 
hinge  scale  h  for  combinations  of  collinear  diagrams 
applies  equally  well  in  such  cases.  (See  Problem  16  of 
Chapter  III.)  Often  a  simple  diagram  may  be 
combined  with  a  collinear  diagram  so  that  the  scale 
on  one  of  the  axes  of  the  simple  diagram  serves 
also  as  the  h  scale  of  the  collinear  diagram.  When  two 
simple  diagrams  are  placed  "back  to  back"  or  super- 
imposed, of  course  no  hinge  scale  is  used. 

20.  Diagrams  with  Parallel  or  Perpendicular 
Indices. — Such     diagrams     consist     of     four     scales 


arranged  in  pairs  corresponding  to  the  four  variables 
of  an  equation  or  formula.  The  scales  are  so  disposed 
that  a  straight  line  drawn  through  a  known  point 
on  a  third  scale  parallel  or  perpendicular  to  a  line 
joining  two  known  points  on  two  other  scales  will  cut 
the  fourth  scale  in  the  point  inscribed  with  the  value 


of    the    unknown    variable.     The    geometric    theory 
involved  is  as  follows:     The  equation 

yo  -  y.  _  b 

Xi  —  .Vi        a 
expresses  the  equaUty  of  the  slopes  of  the  two  lines 
PiPi  and  OP  respectively  as  shown  in  Fig.  66. 
This  equation  above  may  be  written  in  the  determi- 
nant form 


and  another  equation 


a 

b 

0 

Xi 

?'i 

1 

Xi 

}'- 

1 

a 

b 

0 

Xs 

>'3 

1 

Xi 

A'4 

1 

=  0 


regarded  as  a  simultaneous  equation  would  then 
express  the  fact  that  the  lines  PiPo  and  P^Pi  were 
parallel.  If  a  and  b  are  eliminated  from  the  above 
two  determinant  equations  there  results 

^2  —  yi      yu  —  ys 

Xi  —  Xi       Xi  —  Xs 
Consider  now  an  equation  in  four  variables  which  has 
the  form 


^2 


^4    -    ^3 


(41) 


/2    -  /,  fi-  U 

This  equation  may  be  regarded  as  the  result  of  elim- 
inating h  from  the  two  determinant  equations 

1  A  0     I 

.A        g.        1 


h         0 

g2  1 


0(42) 


\i 

Consequently  the  straight  line  index  of  a  diagram  with 
the  defining  equations 


=  /2 


will  be  parallel  to  the  index  of  a  diagram  plotted  with 
the  same  coordinate  axes  and  with  the  defining 
equations 

X  =  fi  y  =  g3 

x  =  h  y  =  gi 

since  both  indices  will  have  the  same  slope  whenever 
Equation  (41)  or  the  equivalent  system  (42)  is  satis- 
fied by  a  set  of  values  of  the  variables  Zi-  .  .  .Zi. 
Example  44. — Lame's  formula  for  thick  cylinders  may 
be  arranged  to  afford  an  example  of  the  use  of  diagrams 
with  parallel  indices.     As  usually  given  the  formula  is 


=  .J 


S  +  P 


IS  -  P 
where  the  letters  have  the  meanings  given  on  the  dia- 


DIAGRAMS  OF  ALIGNMENT  WITH  TWO  OR  MORE  INDICES 


81 


gram  for  the  formula  shown  in  Fig.  67. 

The  formula  may  be  written 

S  +  P  _  D'-  0 
S  -  P  ~   0+  d"" 


Diagram  for 
Lame 's  Formula  for  Thick  Ojlihders 

D=  External  Diamef-er 

d-  Internal        " 

5-  Stress  in  Inner  Surface 

P^ Infernal  Pressure 

Dand  d  m  I  he  iame  uni'fs 

Sand  Pin  f he  same  iinlhs 

A  line  from  dlo  D  Ji  parallel  lo  ahne  from  PfaS 


Values  of  d 


may  at  once  be  written  without  reference  to  the  first 
rows  involving  the  auxiliary  variable  h  but  usually 
scale  factors  will  be  needed  and  since  the  parallelism 
of  all  lines  must  be  preserved  the  scale  factors  for 
\0 


and  may  be  regarded  as  the  result  of  eliminating  // 
from  the  two  simultaneous  equations 


1  1 


1 

h 

0 

p 

-p 

1 

s 

s 

1 

1 

h 

0 

d- 

0 

1 

0 

Z>2 

1 

=  0 


A  set  of  defining  equations  for  both  these  equations 


defining  equations  of  the  second  set  must  be  propor- 
tional to  those  of  the  first  set  thus 


where 


The  method  used  in  this  example  is  general.  The 
underlying  principle  is  the  use  of  a  projective  trans- 
formation that  preserves  parallelism.  (See  Appendix 
B.) 


hp 

y  = 

-y^p 

.r  = 

-M^ 

y=0 

8,S 

y  = 

t^xS 
Ml 

X  = 

M-2 

0 

y  =  i^iD- 

DESIGiX  OF  DIAGRAMS  FOR  EXGIX BERING  FORMULAS 


Diagram  for 
Lams' ^6  Formula  for  Thick  Qj finders 

D= Exhrnal  Dia  meter 

ct=  Infernal  Diamefer 

S'Sfress  In  Inner  Surface 

P'  Internal  Pre&sure 
D  and  d  in  the  same  anih 
Sand  Pin  the  same  unlls 


A  line  fromdlo  D is  perpendicular  toa  line  from  PfoS 


DIAGRAMS  OF  ALIGNMENT  WITH  TWO  OR  MORE  INDICES 


In  the  present  example  the  scales  are  all  straight 
and  readily  plotted.  The  same  units  must  be  used 
for  5  and  P  such  as  tons  or  thousands  of  pounds: 
also  in  using  the  D  and  d  scales  the  same  units  must  be 
employed,  as  inches  or  centimeters.  The  indices  are 
shown  set  for  P  =  4,000,  5  =  10,000,  d^  Q  required 
D.  Reading  of  the  diagram  may  sometimes  be 
made  more  convenient  by  providing  a  transparent 
piece  of  celluloid  on  which  parallel  lines  are 
scratched. 

It  is  now  quite  evident  that  Equation  (39)  of  Article 
18  may  be  represented  also  by  a  diagram  with  parallel 
indices.  In  fact  Equations  (40)  constitute  the  necessary 
reduced  determinant  equations.  This  Equation  (39) 
serves  also  to  show  that  where  the  scales  of  Equation 
(42)  reduce  to  straight  scales  supported  respectively 
in  pairs  on  the  same  straight  lines,  the  necessary  theory 
of  the  parallel  alignment  of  the  indices  is  merely  the 
geometry  of  similar  triangles.  In  case  the  supports 
of  the  scales  are  parallel  the  segments  intercepted  by 
the  two  indices  are  proportional.  Those  who  are 
familiar  with  the  use  of  homogeneous  coordinates  in 
geometry  will  recognize  that  the  presence  of  zero  in 
place  of  unity  in  the  third  element  position  of  the 
determinants  of  Equations  (42)  merely  indicates  that 
the  resulting  diagram  with  parallel  indices  is  a  special 
case  of  the  diagrams  of  double  collineation  in  which 
the  hinge  scale  has  been  removed  to  infinity. 

It  is  possible  to  give  equations  of  the  form  (42), 
which  includes  Equation  (40)  as  a  special  case,  another 
simple  diagrammatic  representation.  In  this  repre- 
sentation the  key  to  the  solution  is  by  perpendicular 
indices  instead  of  by  parallel  indices  and  it  has  some 
advantage  because  of  the  fact  that  two  perpendicular 
lines  scratched  on  a  piece  of  transparent  celluloid 
wUl  serve  as  the  two  perpendicular  indices  and  both 
pairs  of  scales  may  be  read  at  one  setting.  Bearing 
in  mind  that  It  in  both  determinants  of  Equation  (42) 
represents  the  variable  slope  of  the  two  indices  it  is 
only  necessary  to  replace  unity  in  the  first  element 
position  of  the  second  (or  first)  determinant  by  —h, 
and  h  in  the  second  element  position  of  the  first  row 
by  unity  in  order  that  the  slopes  of  the  two  indices 
shall  be  no  longer  equal  but  one  the  negative  reciprocal 
of  the  other  whenever  they  are  to  determine  cor- 
responding values  of  the  four  variables  Zi  .  .  .  Z4. 
The  corresponding  change  in  Equation  (41)  requires 
that  equation  to  be  written 


gi 


h-h 


gi-  gi 


(41a) 


but   the   defining   equations    are   selected   from    this 
equation  exactly  as  they  were  for  Equation  (41).     In 


other  words  the  original  Equation  (41)  may  be  repre- 
sented by  a  diagram  with  perpendicular  indices  by 
writing  first  the  two  determinant  equations 


=  0 


1 

h 

0 

gl 
g-i 

1 
1 

1 

1 

h 

gi 

-h 

gi 

-u 

(42a) 


as  a  check  and  constructing  the  diagram  from  the 
defining  equations 


/l 

y  =  gi 

x  =  gz 

y  =  -/s 

72 

y  =  g2 

X=   gi 

y  =  -U 

Example  45. — From  the  formula  of  the  preceding 
example  another  set  of  defining  equations  may  be 
written  which  will  yield  a  diagram  with  perpendicular 
indices  as  follows 


byP 

biS 


biD^ 
0 


y  =  0 

y  =  md^ 


and  the  diagram  is  shown  in  Fig.  68. 

21.  Diagrams  for  the  Equation  /i  -f  /o  .  •  .  + 
/„  =  0.  — Sometimes  the  Equation  (38)  in  Article 
19  has  the  simple  form 


/>+/.= /a +/4 


(43) 


and  a  diagram  of  double  collineation  with  parallel 
straight  scales  may  be  constructed  by  using  as  before 
the  auxiliary  variable  h  and  the  two  equivalent 
equations 


h^h-h 


h  +  U 


The  corresponding  determining  equations  with  suitable 
scale  factors  will  then  be  (Article  12,  Chapter  III) 


x=  -5i 

y  =  Mi/i 

X  =  - 

63 

y  =  /is/a 

X=           b2 

y  =  M2/2 

X  = 

«4 

.V  =   M4/4 

X  =      0 

y  -  -     ,    ■  ■ 
m  -f-  iJi 

X  = 

0 

Ma  +  /i4 

where  it 

is  necessary  in 

addition  to  the  conditions 

imposed 

upon  the  constants  m  and  b 

in 

Article  12  to 

require  a 

so  that 

MiM2 

M.  +  M2 

M3M4 

Ala  +  M4 

(44) 

in  order  that  the  same  value  of  the  auxiliary  variable 
h  shall  always  determine  the  same  point  on  the  hinge 
scale.  The  scheme  of  the  resulting  diagram  is  shown 
in  Fig.  69  and  Example  42  illustrated  its  application. 


84 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


If  it  is  found  more  convenient  the  determinant 
equations  used  for  the  Equation  (43)  may  be  written 


2 

-/i 

/2 
h 

2 

/4 


0 

Fig.  69. 


The 


h 

V  = 

-M.A 

X  = 

-82 

y  = 

-M-./3 

Si 

>'  = 

M./2 

x  = 

&2 

y  = 

/X2/. 

and  a  diagram  with  parallel  indices  designed, 
defining  equations  with  scale  factors  are 


where 


If  5i  =  62  then  of  necessity  mi  =  ^2,  and  the  supports 
of  the  scales  coincide  in  pairs  and  the  resulting  diagram 
is  shown  in  Fig.  70. 


f,\ 

/3 

JL^'-""". 

5 

'"'" 

■ 

- 

~- 

New  types  of  diagrams  may  now  be  constructed 
for  the  equation  (7)  of  n  variables 


Introduce  the  auxiliary  variables  h\,  hi,   .    . 
and  write  the  equivalent  system  of  equations 

/l+/2-/^l  =  0 
hx  +  /3  -  //2  =  0 
hi   +  /4   -    //3    =    0 


^„-3   +/„-!+/„=    0 

In  each  of  these  equations  except  the  first  one  and  the 
last  one  two  auxiliary  variables  h  enter  and  one  value 
of  h  must  always  have  a  support  for  each  application 
of  the  index  in  the  diagram.  For  example  h-i  may  be 
determined  from  the  first  two  equations  written  in  the 
form 


1 

h, 

2 

0 

1 

-h 

1 

1 

h 

1 

1 

h, 
2 

0 

1 

u 

1 

1 

hi 

1 

and  represented  by  a  corresponding  diagram  with 
parallel  indices  in  which  no  support  appears  for  h\ 
but  in  which  one  does  appear  for  hi.  Figure  71  shows 
the  scheme  of  such  a  diagram. 


f, 

f 

h        '■- 

/ 

; 

5-                  / 

i        h     h     -V           fe 
-   ~  7''"' 

/1  +  /2+/3  + 


+/» 


(7) 


It  is  not  necessary  of  course  to  use  the  principle  of 
parallel  indices  to  construct  the  diagram  for  Equation 
(7)  as  hinge  scales  can  be  used  throughout,  but  is 
frequently  convenient  to  do  so.  The  spacing  of  the 
scales  and  the  use  of  the  scale  factors  are  controlled  at 
each  step  by  principles  already  laid  down  in  this  chap- 
ter and  in  Chapter  III. 

Example  46. — The  formula 

JFD 

for  determining  the  actual  time  for  turning  a  piece  of 
work  in  a  lathe  is  shown  in  Fig.  72  and  the  symbols  of 
the  formula  are  described  on  the  figure.  If  written 
log  r  +  log  5  -  log  0.2618  -  log  /^  -  log  L 

-  log  Z?  =  0 


DIAGRAMS  OF  ALIGNMENT  WITH  TWO  OR  MORE  INDICES 


the  formula  is  in  a  form  similar  to  Equation  (7).     If  between  the  scales  for  D  and  L.     Rearranging  it  in 

P,  the  product  LD  is  introduced  as  an  auxiliary  variable  the  form 

the  equations  are  log  P  —  log  Z,  =  log  D 

log  r  +  log  5  =  log  k  =  \og  F  +  log  P  +  log  0.2618  results  in  placing  the  D  scale  between  the  other  two 

log  P  =  log  L  -\-  log  D  scales  thus  permitting  the  auxiliary  diagram  for  DL  = 


I- 


W      60- 


3 


3 


-     -  g 


UiS 


The    first    equation  yields    the    following    defining 
equations 

X  =  —26  y  =  fjilogF 

x=      25  >>  =  M  log  P  +  M  log  0.2618 


C  50^  <7} 


;§ 


Is 


T  -8 


E'5 

—  6    S 
-53 


Fig.  72.— Diagram  for  T  =  0.2618  ^  t!. 

P  to  be  joined  conveniently  at  the  side  of  the  diagram 
for  the  first  equation  instead  of  superimposing  it 
upon  the  latter. 

The  defining  equations  for  the  second  equation 
referred  to  an  origin  on  the  Z>  scale  are 


log  5 
logr 


The  equation  P  =  LD  if  plotted  from  its  form  as  given 
above,  would  require  that  the  scale  for  P  be  located 


x=  -Si 
x=  0 
X  =      5i 


y  =  fi  log  P 
>-  =  glogZ? 
y  =  i^logL 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


Zl\ 


"m^W. 


lYtl 


/// 


a.o-  -i>» 
1.8^  -IK 


■g  ^  -s 

So.sL  -2 

■5  r  15 

^    ■■  a 

5-  ^ 


VulbMofFactorfl/) 

y 

y 

llih 

L>».lul. 

C."il 

^ 

■'To"" 

^:e 

IJ 

0  078 

0.007 

0  052 

37 

0  111 

0  100 

OOOl 

0.058 

30 

0102 

oes 

u 

IS 

0.076 

38 

0.122 

067 

<3 

>' 

0.080 

0.130 

110 

070 

10 

0.050 

0.103 

0.000 

100 

0.1«2 

0.113 

IS 

0.108 

0.087 

0.003 

~'  1  0.'5« 

0.12. 

.,.W6 

The  Lewis  formula  for  the  strength  of  gear  teeth  is 

ty^SPby;        or,  if  h  =KP;        W=SPtKy. 
A  line  from  S  \o  ^V  crosses  the  center  line  at  the 
point  as  a  line  from  A"  to  P. 

EXAMPLE. 
Given:  Cast  Iron  gear,  velocity  of  teeth  =150  ft.  per 
W  =  3SOO  lbs.  "load  carried  by  teeth. 
^=0.11. 
A- =4. 
RequiredJ    P 

As  shown  by  dotted  lines,  P  >-l.(Me 


DIAGRAMS  OF  ALIGNMENT  WITH  TWO  OR  MORE  INDICES 


87 


Example  47. — In  the  Lewis  formula  for  the  strength 
of  gear  teeth 

W  =  SPb{y) 
it  is  often  desired  to  solve  for  P  if  &  is  taken  as  KP, 
where 

W  =  load  in  pounds  carried  by  the  teeth 
S  =  stress  in  pounds  per  square  inch,  chosen  with 
reference  to  the  velocity  and  material  of  the 
teeth 
P  =  circular  pitch  in  inches 
b  =  width  of  face  in  inches 
K  =  a,  constant,  usually  2  to  6 
(y)  =  a   factor   depending   upon   the   number  and 
shape  of  the  teeth. 
In  logarithmic  form  the  formtila  becomes 
log   IF  -  log  5-2  log  P  -  log  K-  log  (y)  =  0 
but  if  the  auxiliary  variable  hi  is  chosen  so  that 

it  becomes 

log  hy  -  log  5  =  log  /?  =  2  log  P  +  log  K 
Figure  73  was  then  plotted  from  the  defining  equations 

W 

X  =  -2h  y  =      li  log  hi  =  n  log- 

x=      23  y  =  -^logS 


■iy) 


-5 
8 


'logh 


y=      2i,  log  P 

y  =        filogK 
The  first  two  equations  define  a  binary  scale  on  the 
line  X  =  —25.     The  curve  net  drawn  for  this  scale 
may  consist  of  the  lines 

X  =  -iy),  y  =  Mlog|^3^J 

plotted  with   the  line   x  =  —28   as  a   new   Y  axis. 
(See  Article  16,  Chapter  IV.) 

From  a  table  given  by  Mr.  Lewis  the  values  of  the 
velocity  were  added  in  proper  correspondence  with 
the  scale  for  S.  Since  the  product  of  the  diametral 
pitch  and  the  circular  pitch  is  always  t  a  scale  for  the 
diametral  pitch  was  added  to  the  P  scale. 


The  equation  hi 


could  naturally  be  written 


in  the  logarithmic  form  and  two  auxiliary  (collinear) 
logarithmic  scales  for  IF  and  (y)  added  to  the  present 
figure  just  as  was  done  for  L  and  D  in  Fig.  72  for  the 


preceding  example.  The  range  of  numbers  for  (y) ,  how- 
ever, is  very  small  and  it  was  found  more  convenient 
to  use  the  system  of  curves  and  establish  a  binary 
scale  on  the  line  x  =  —25  as  shown  in  the  figure. 

Problem  1. — Discuss  the  equation  of  Example  47  as  an 
equation  of  type  (7)  and  construct  the  diagram  resulting 
when  the  binary  scale  is  replaced  by  the  required  parallel 
scales  for  IF  and  (y)- 

Problem  2. — Reduce  Bazin's  equation  for  the  velocity  of 
water  to  type  (7)  and  construct  a  corresponding  diagram. 

Problem  3. — Construct  a  practical  diagram  for  the 
formula  of  E.xample  12  of  Chapter  II.  Write  useful 
values  of  c  and  use  four  parallel  lines. 

Problem  4. — Gordon's  formula  for  columns  is 


1  + 


5a 

600^2 


where  W  =  safe  unit  load,  2,725  to  14,450 
a  =  coefficient,  2,000  to  3,200 
I  =  unsupported  length  in  inches 
d  =  least  dimension  in  inches 
I 
d 
If  the  determinant  equation  for  I  and  d  is 

1  h  0 

1  -  600d^ 


8  to  40. 


0 


600  d^ 
1  +  P  -  1 

P  P 

find  the  corresponding  determinant  equations  for  a  and 
PF  and  design  a  diagram  of  parallel  alignment  with  suit- 
able scale  factors  for  practical  use  in  steel  design. 

Problem  5. — In  Section  15  of  Chapter  III  were  described 
diagrams  of  alignment  with  a  fixed  point;  show  that 
Equation  (41)  can  be  represented  by  a  diagram  with  the 
fixed  point  x  =  I,  y  =  0  and  two  binary  scales  on  the  Y 
axis. 

Problem  6. — A  reduction  formula  used  in  automobile 
radiation  tests  is 

62.4^1 


H2 


1  + 


0.24^ 


where  A 
Di 
Hi 


OASAJ), 

W     "^      H\ 
(6,000  to  15,000)  air,  pounds  per  hour. 
(80°  to  115°F)  mean  temperature  difference 
(40,000  to  90,000)  heat  transfer  observed, 
B.t.u. 
IF  =  (1,000  to  3,500)  water,  pounds  per  hour. 
Show  how  to  design  a  diagram  of  double  alignment  with 
parallel  scales  for  this  formula  and  with  the  quantities 

grouped  in  the  pairs  H,A,  and  W,  K  where  K  =  ^' 


CHAPTER  VI 


ALIGNMENT    DIAGRAMS    WITH    ADJUSTMENT 


Introduction. — There  is  introduced  in  this  chapter 
a  new  class  of  diagrams  based  upon  fundamental 
principles  already  developed.  These  diagrams 
enlarge  the  number  of  types  of  equations  to  which  the 
principle  of  collineation  is  immediately  applicable 
and  furnish  also  a  general  alternative  method  espe- 
cially for  those  equations  which  cannot  readily  be  iden- 
tified with  preceding  types.  It  will  moreover  be 
found  that  the  types  of  equations  already  treated 
may  be  regarded  as  special  cases  of  the  more  general 
types  now  to  be  discussed. 

22.  Equations  in  Three  Variables. — It  was  shown 
in  Chapter  III  that  an  equation  in  three  variables 


/i 


0 


may  be  represented  by  a  collinear  diagram  with  three 
scales  when  and  only  when  it  can  be  written  in  the 
reduced  determinant  form 


/3  g.  1 


=  0 


(8) 


Since  there  is  no  immediate  and  satisfactory  test  for 
this  desired  property  of  an  equation  or  formula  it  is 
usually  necessary  to  resort  to  the  principle  of  compari- 
son with  certain  type  forms  and  to  the  tentative  rule 
of  Article  14.  It  is  therefore  desirable  to  free  the 
determinant  form  from  restrictions  as  far  as  possible 
and  at  the  same  time  to  preserve  the  principle  of 
collineation,  or  the  use  of  the  straight  line  index  in 
designing  the  diagram. 

The  fundamental  property  of  Equation  (8)  is  the 
segregation  of  the  variables /i, /o, /a,  into  rows  of  the 
determinant. 

Consider  now  the  determinant  equation  of  the  type 


/l2  gl2  1 

/•23  g-23  1 

fu  ^31  1 


=  0 


(45) 


in  which  the  elements  of  each  row  are  allowed  to  con- 
tain at  most  two  variables,  which  variables  may  appear 
in  more  than  one  row. 


Then  the  equations  analogous  to  the  previously 
defined  and  much  used  defining  equations  will  be  of 
the  type 

x=fi2  y  =  gi2 

X  =f23  y  =  g23  (46) 

X  =  fii  y  =  g3i 

and  each  pair  of  equations  will  determine  a  curve 
net,  except  in  the  case  explained  below  in  Article  23. 
The  three  curve  nets  are  shown  schematically  in  Fig. 
74  and  it  is  seen  that  there  appear  two  families  of 
curves  for  each  variable  z. 


Call  any  set  of  three  values  of  z  which  simultane- 
ously satisfy  Equation  (45)  corresponding  values  of  z. 
Such  values  of  z  necessarily  determine  values  of  the 
functions  /  and  g  and  hence  by  Equations  (46)  there 
result  three  pairs  of  coordinates  x  and  y  which  must 
satisfy  Equation  (45)  when  substituted  for/ and  g. 
But  Equation  (45)  would  then  express  the  geometric 
fact  that  three  points  in  the  three  respective  curve 
nets  are  collinear. 

In  general,  however,  it  would  not  be  true  that 
any  three  points  taken  at  random  in  the  three 
plotted  curve  nets  and  on  the  same  straight  line 
would  yield  corresponding  values  of  z  attached  to 
the  curves  intersecting  in  pairs  at  the  respective 
points.  It  is  here  that  the  present  theory  departs 
from  the  theory  previously  developed.  What  happens 
in  general  is  that  there  appear  six  values  of  z  consisting 
of  three  pairs  of  dissimilar  values. 

When  corresponding  values  of  z  are  used  to  select 
three  points  in  the  three  curve  nets  it  is  seen  at  once 
that  the  same  value  of  z  is  used  to  select  a  curve  from 
two  different  nets.     If  now  one  variable  Zo  is  unknown 


ALIGNMENT  DIAGRAMS  WITH  ADJUSTMENT 


=  0 


(47) 


it  is  evident  that  the  index  must  be  rotated  about  the 
point  always  determined  in  one  net  by  the  known 
values  ZiZz  until  the  same  value  of  the  unknown  z-i 
appears  in  the  two  remaining  nets  at  the  points  of 
intersection  of  the  index  with  the  known  curves  in 
each  net.  In  the  iigure  the  line  PR  is  rotated  about 
R  until  the  points  of  intersection  P  and  Q  determine 
the  same  value  of  the  variable  z^  when  it  is  assumed 
that  Zi  and  Zs  are  known. 

This  then  is  the  principle  of  collinear  diagrams  with 
adjustment.  There  are  many  special  cases  and  in  not 
a  few  no  adjustment  of  the  index  is  required  because 
the  unknown  variable  appears  but  once.  The  method 
is  of  great  practical  advantage  especially  if  a  given 
equation  is  not  adapted  to  the  preceding  treatment. 

23.  Special  Forms  of  Equations.— Equation  (45)  is 
a  general  form  and  is  less  frequent  than  the  simpler 
special  cases.  For  example  /31  and  gzi  may  often 
reduce  to  fz  and  gz  respectively  by  skillful  choice  of 
the  elements  of  the  determinant.  The  corresponding 
determinant  equation  is  then 

/12  gl2  1 

fiZ  g2Z  1 

A  simpler  form  of  diagram  results  from  this  equation. 
Without  scale  factors  the  defining  equations  are 

X  =  fi2  y  =  gi2 

X=  fiZ  y   =    g23 

x=Jz  y  =  g3 

The  first  pair  of  equations  lead  to  the  curve  net, 

F,{xy)  =  Zi  Fiixy)  =  Zi 

Similarly  from  the  second  pair  is  obtained  the  curve 
net 

Giixy)  =  Z2  Gz{xy)  =  Zz 

and  the  third  pair  of  equations  determine  a  curved 
scale  for  z  with  the  support 

S(xy)  =  0 
Still  more  simple  is  the  equation 

/2  g2  1 

/23  g23  1 

In  the  resulting  diagram  there  wil 
a  curve  net  defined  as  follows 

x=f, 

x  =  f, 

X  =  hz 

Frequently  a  redundancy  of  variables  in  an  equa- 
tion may  be  reduced  by  the  introduction  of  a  param- 
eter which  is  a  simple  function  of  two  or  more  of  the 
variables  whose  values  are  always  given,  as  was  done 
in  Chapter  III  in  the  case  of  Example  33  for  the  mean 


=  0 

(48) 

ill  be  two  scales  and 

y  = 

gi 

y  = 

g-2 

y  = 

g2  3 

hydraulic  radius  of  trapezoids.  This  device  will 
be  of  advantage  in  several  of  the  examples  which 
follow. 

Example  48.- — As  a  first  illustrative  example  of  the 
Equation  (48)  the  quadratic  equation 

z^  +  aiZ  -{-  a-i  =  Q 
may  be  written  in  the  reduced  determinant  form 
-a,        0  1 

0         2  1=0 

z  k  1 

where  k  = is  a  parameter.     The  three  variables 

are  ai,  2,  and  k.     The  defining  equations  are 
x  =  —ai  y  =  0 

X  =      0  y  =  z 

X  =       Z  y  =  k 

To  solve  a  quadratic  equation  by  this  diagram  the 
figure  is  entered  on  the  X  axis  with  the  value  of  ai  at 
P  and  the  index  is  then  turned  about  this  point  until 
the  value  of  2  read  on  the  Y  axis  is  the  same  as  the 
value  read  on  the  vertical  line  intersecting  the  index 
aj 
ai 
is  set  for  the  two  roots  of  the  equation 

z^  -  6.2z  -  18.6  =  0 

Another  simple  case  of  equation  (45)  is 

/12         gu         1 

fz  gz  1        =0 

fi'      gi       1 

Example  49. — As  an  illustration  of  this  form  of  an 
equation,  the  equation 

21Z2  -  23  +  Vl  +  Z2-VI  +  sr  =  0 
which  can  not  by  algebraic  transformation  be  given 
the  form  of  Equation  (8),  may  now  be  considered.     It 
may  at  once  be  written  in  the  first  determinant  form 
1  0 


where  it  is  crossed  by  y 


In  Fig.  75  the  index 


(49) 


0 


=  0 


1  Vl+Zi 

Zi  Vl   +  22-  Z3 

from  which  by  adding  columns  one  and  two  for  a  new 
second  column  and  then  interchanging  columns  two 
and  three  and  dividing  by  the  elements  of  column 
three  there  results  the  reduced  determinant  equation 

1  Zy  1 


Vl+Zl 
Z3 


=:-=„       1 


S2   +    Vl    +  Z2-        Z2    -1-    VI    +   22  = 

For  a  good  arrangement  of  the  drawing  the  vertical 


90 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


unit  may  be  taken  one-tenth  of  the  horizontal  unit  and      equations  leads  to  the  equation  of  a  family  of  parabolas 
there  follow  the  defining  equations  2  _  ^^  n  _      ^ 

2j  for  segregating  the  values  of  Z3.     The  finished  diagram 

y  ~  u\  is  shown  in  Fig.  76. 

In  constructing  diagrams  of  adjustment  it  is  desir- 
able if  possible  to  avoid  binary  scales  as  the  use  of  the 
10  diagram  is  complicated  by  their  presence. 


10 


10          ' 

8 

.           2 

14^56 

-i:.: 3: - 

|I|M||||H]|I|| 

Jl 

I 

:  ^ 

r     -1^1-7 

q+ , 

qq::;:::: 

:::::::  it::::::::::::::::::: 

1                  III 

"F::t:::x:::: 
-r +rTT — r--i 

.  \m 

::::::::::!j:3::::::::::::±:::: 

1               1       I'M 

-^'r4^--p^---L- 

:.c 

.1 _^ -  -- 

-  4 

i ' 

1           J 

h^-t  T 4-i 

- 

'  ''i^' 

i±:::::: 
,  1 

i:i=|::::::::i:::g::ii::::: 

ij 

--4?- 

::::::::::ii^^^:: 

HiS::::::-:T;:-:::::::::x:::::: 

1^4:;;- 

:  2 

fEMMttm™ 

::^r:::::::::: 

:  ' 

■;;;;|-Sife;a.. 

H--%f:^  =  =  -5s44^--i^M 

a 

:::::l::ii;p.::::l.; 

::::::::::lSS:::::::l-it::: 

-t-i- -4.1              -    .    .             i 

'i 

'"    !  ■,  r-;   ''" '  ■  I    ": ;  : 

3;;^x;::I&:::S::::;!|H 

"1  — 

=;r^^"^"~^^7rT 

T^^^^'^sM:^^ 

^t 

0,    :::::: 

--^Xi^px     '  '      '    ''MI- 

|f:|:::::::,,^_:::±^:^:::::::: 

|^;:::;:::::!::;::|::ili 

::::X:-i:':^:::: 
1  1  1  -j-.- 

i' 

m 

rt':::::::::::::  ::::::::::::::: 
tt  ::::::::::::::::   :::::::::::::: 

is 

:  :::i 

mm 

-  - +f  44+h +f++ +rf+ 4-W- -l-l-i- - -1- 

rflifiMfflfflMIIIIIIIIIIIII 

: 

9 

8            1 

6            5^ 

2     J        ^ 

1            8            S 

Example  50. — The  quadratic  equation 

z'^  -f  aiZ  -f  a2  =  0 


22  +  VI  +  z^  10  (z2  +  Vl  +  22') 

The  two  scales  for  Zi  are  easily  drawn.     For  the      may  be  written  in  the  form 
curve  net  for  Z2,  Z3,  since  the  right  side  of  the  first 
equation  involves  only  Zi,  there  result  straight  lines 
parallel  to  the  Y  axis  for  the  curves  of  that  variable. 
The  elimination  of  the  variable  Zi  between  the  last  two 


-1 

aiZ 

1 

0 

-z' 
2 

1 

1 

02 

1 

ALIGNMENT  DIAGRAMS  WITH  ADJUSTMENT 


91 


which  is  another  special  case  of  Equation  (45) .  With 
a  horizontal  unit  twice  the  vertical  the  defining  equa- 
tions become 

X  =  —2  y  =  Ci2 

—  2^ 

a;=      0  3'=  ^ 

x  =      2  y  =  ai 


appear  at  Q  and  R  and  the  operation  is  somewhat 
difficult  to  manage.  In  the  figure  the  index  is  set 
twice  for  the  roots  of  the  equation 

z^  -  0.8z  -  6.6  =  0 

It  will  be  seen  that  whenever  any  variable  z  appears 
in  but  one  row  of  the  determinant  equation  of  the 


2.-S     Z^.J0S_Z^6_ 


A  binary  scale  is  required  on  the  line  x  =  —  2  and 
the  variables  ai  and  z  may  be  segregated  by  setting 

y  =  zx  X  =  ai 

the  resulting  diagram  is  shown  in  Fig.  77. 

The  roots  are  determined  by  turning  the  index  about 
the  point  P  on  the  aj  scale  until  the  same  values  of  z 


form  (45) ,  no  adjustment  of  the  index  is  necessary  in 

determining  its  value  from  the  corresponding  diagram. 

It  is  to  be  noticed  that  the  successive  elimination 

of    two    variables   Zi    and   Z2    from    two    equations 

X  =/i2,  y  =  gu, 
will   fail  if   the  two  functions  fu  and  gu  are  not 


DESIGN  OF  DIAGRAMS  FOR  EXGIXEERIXG  FORMULAS 


independent  functions;  that  is  to  say  in  case  one 
is  a  function  of  the  other.  Sometimes  this  con- 
dition plainly  arises  because  both  functions  are 
functions  of  the  same  combination  of  the  two  vari- 
ables z. 


results  always  the  equation  of  a  curve  which  is  the 
curve  support  for  a  binary  scale. 

In  the  above  example  it  is  seen  that  to  every  point 
of  the  parabola  there  corresponds  an  indefinite  number 
of  pairs  of  values  z^z-i  and  to  segregate  them  either 


For  example,   suppose   that 

/i2  =  ZiZo   and  gn  =  VziZq 
then   it   is   evident   that   both   variables   are   elimi- 
nated simultaneously  from  /12  and  gn  and  that  there 
results  the  parabola  y^  =  x.     Whenever  both  variables 
are    eliminated    simultaneously   in    this   way    there 


one  of  the  defining  equations  may  be  used.  Choosing 
X  =  ZxZi,  any  simple  family  of  cur^'es,  except  the 
lines  parallel  to  the  Y  axis,  may  be  selected  to  define 
one  of  the  variables,  say  y  =  Zi ;  substituting  this 
value  of  Zx  in  the  last  equation  yields 
X  =  yzi 


ALIGNMENT  DIAGRAMS  WITH  ADJUSTMENT 


and  all  lines  of  the  two  systems  which  intersect  on 
the  same  ordinate  {x  =  ZiZ->^  determine  pairs  of  values 
of  Z\  and  z<i  which  correspond. 

However,  in  using  the  diagram,  the  index  must 
always  pass  through  the  point  P  in  which  the  parabola 
is  cut  by  the  ordinate. 

It  will  be  noticed  that  a  similar  segregation  of  the 
variables  could  have  been  obtained  by  starting  with 


frequently   arise.     For  example,   one   variable   may 
enter  every  row  of  the  determinant  thus 


(50) 


and  when  Z\  is  known  no  adjustment  of  the  index 
is  required. 


/n 

gl2 

1 

/l3 

gl3 

1 

/i 

^1 

1 

the  second  defining  equation  y  =  -s/z^z^  and  selecting 
an  arbitrary  curve  net  for  either  Zi  or  Zi  and  then  the 
corresponding  values  of  2i  and  Zi  would  have  been 
found  on  the  curves  in  the  resulting  net  which  intersect 
on  the  same  abscissa,  or  line  parallel  to  the  axis  of  X. 
The  scheme  of  the  curved  binary  scale  discussed  here 
is  shown  in  Fig.  78. 

24.  General  Forms  of  the  Equation  in  Three  Vari- 
ables.— Equations  closely  allied  to  Equation  (45)  will 


Equations  (45),  (47),  (48),  (49)  and  (50)  are  special 
cases  of  the  equation  of  more  general  form 


(51) 


where  the  subscripts  z'.j,  etc.,  are  allowed  to  take  on 
in  pairs  any  values  from  the  set  of  numbers  0,  1,  2,  3 
with  the  understanding  that  0  shall  denote  the  absence 


u- 

gii 

1 

u 

gkl 

1 

^mn 

g^n 

1 

94 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


of  a  second  variable  in  any  function  in  which  it  is 
written.  There  are  eight  distinct  types  of  equation 
(51)  if  all  trivial  cases  are  excluded.  These  eight 
types  have  the  following  determinants  for  the  left 
hand  member: 


I 

I/.        g. 

1/3  g3 


: ' .!;: 


II  III 

«i  1  I  I  /.  «.  1 

g2  1   U  /i2  gn  1 

/l3  gl3  ill   /l3  ^13  1 


IV  V  VI 

I  /i      gi      1     [  M     gu     1  !  I  /.      g.      1 
\m     gi2     1   ,  /u     su     1  '.  /23     g23     1 

I    /"23  g23  1  I   hi  gii  1,1   f'n         g'23         1 

VII  VIII 

I    /.2  gl2  111   -^^  ^'  ^ 

I    /l3  gl3  1     j,   I    A  gl  1 

/'l3         g'l3  1     j     1   /23  g23  1 

It  is  seen  that  in  cases  III  and  VII  it  is  possible  that 
the  unknown  variable  may  enter  in  each  row  in  which 
case  no  initial  fixed  point  would  be  determined  and  the 
position  of  the  index  which  yields  a  solution  would 
only  be  found  by  trial  and  error.  On  the  other  hand 
in  these  same  two  cases  if  the  variable  entering  each 
row  is  known  there  is  no  adjustment  of  the  index 
required,  unless  S3  is  unknown  in  VII.  Nor  is  adjust- 
ment required  in  several  other  cases,  e.g.,  II  when  23 
is  unknown,  etc.  Case  I  is  of  course  Equation  (8)  in 
three  variables. 

In  practice  one  (or  more)  of  the  functions  may 
reduce  to  a  constant  in  which  event  a  binary  scale 
is  needed  in  carrying  out  the  construction  unless 
the  other  function  in  the  corresponding  row  con- 
tains but  one  variable;  in  that  case  only  a  straight 
scale  is  involved.  If  the  unknown  value  is  involved 
in  the  net  of  the  binary  scale  as  mentioned  above  the 
management  of  the  index  becomes  troublesome  unless 
a  known  value  enters  in  each  row  of  the  determinant. 

There  are  two  additional  cases  in  which  the  number 
of  variables  reduces  to  two,  viz.: 


/• 

gi 

1 

/l 

gl 

1 

h 

gi 

1 

=  0 

/.2 

gli 

1 

M 

gn 

1 

fu 

g'n 

1 

0 


and  there  is  still  one  other  permissible  case  in  which 
all  the  elements  of  a  row  reduce  to  constants,  for 
example  the  case 

/12  gl2  1        I 

/3  1  11    =    0 

0         0         1    I 
(Compare  this  equation  with  Equation  (16)  Chapter 
III.)     The  above  equation  would  be  represented  by  a 


diagram  with  a  net  of  curves  and  a  straight  scale  on 
the  line  y  =  I  and  the  index  would  always  pass 
through  the  origin.  No  adjustment  is  needed.  If 
/12  and  gi2  should  occur  in  the  form  /i  +  gi  and  /i  — 
g2  respectively,  ordinary  cross-section  paper  could  be 
utilized  to  plot  the  resulting  system  of  perpendicular 
lines  and  the  scale  for  the  function /a  would  appear  on 
a  diagonal  line. 

25.  Equations  in  More  than  Three  Variables. — It  is 
possible  to  construct  diagrams  for  equations  of  the 
type 

.A2  gv2  1        I 

/23  g-2Z  1  =0  (52) 

fsi  g3.  1        I 

In  the  most  general  case  there  will  be  three  nets  of 
curves  and  it  is  to  be  observed  that  whenever  Zi  or 
24  is  unknown,  no  adjustment  of  the  index  is  neces- 
sary. There  are  many  simple  cases  including  the 
examples  of  Chapter  IV. 

Example  51.— The  general  cubic  equation 
2"  4-  ai2=  +  a.,2  +  a3  =  0 
can  be  written  in  the  reduced  determinant  form 


z' 

0  2 

—  02       —a 


02 


The  quotient  of  the  two  coefi6cients  —  may  be  repre- 
sented by  the  parameter  K  and  the  four  variables  are 
then  apparent.  It  is  convenient  to  introduce  the 
scale  factor  2  throughout  the  ordinates  and  the  dia- 
gram may  be  designed  on  a  sheet  20  by  20  inches  as 
shown  in  Fig.  79. 

The  defining  equations  are 

a;  =      2^  y  =  -2K 

x=      0  y  =      2z 

X  =  —Ui  y  =  — 2ai 

The  diagram  must  be  entered  with  the  values  of  02 
and  fli,  then  the  index  is  rotated  about  the  correspond- 
ing point  until  the  value  of  2  on  the  F  axis  is  identical 
with  the  value  found  at  the  point  where  the  index 

crosses  the  hne  y  =  —  2— .     In  the  figure  the  indices 
are  set  for  the  three  real  roots  of  the  equation 
2»  -  0.5  z^  -  7.52  -h  9  =  0 
Example  52.— Another  excellent  example  is  alTorded 
by  the  formula  for  the  length  of  the  belt  connecting 
two  pulleys  and  known  as  the  Open  Belt  Formula 

L=  R(^  +  26)  +  r(ir  -  29)  +  2C  cos  9 
Where  R  and  r  are  the  radii  of  the  two  pulleys  and  C 


ALIGNMENT  DIAGRAMS  WITH  ADJUSTMENT 


95 


the  distance  between  their  shaft  centers.     The  angle      and  the  defining  equations  with  the  horizontal  unit 
e  is  determined  by  the  equation  ten  times  the  vertical  are 

e  =  ,rc.\.?^^^  ^=10  y=.R 


C  may   be  taken  100  as  a  standard  and  R  and  r 
expressed  as  decimal  parts  of  C.     d  is  essentially  a 


■  10 
20 


100  cos  e 


■10              S                :,               7               6                                4               5 

■^:"-;i:;:::;:i:;ii;-gs;;;;|;;;;;;:;r 

4 -J_._ 

::::::::::::::-: ^-T""T""t+t^Trfr  f^^-ri-" 

;j:::5:t:;:l:;H±;i:r:::;;:;±:::::::::::; 

1[[                      H'  Ph'  Ni  '"'I 

_,i..^. — ""1^ 

L^ ^ 1_|„| 

^  -I—      1     \ 

r^^Tii: ■--  - T- 

hhi  1  1              1 

:::::::;  :::::::::::::::il;::;::=t;:::::::;:::x::: 

i__3::+:::::::::::::^:::::::::::::::::::::S 

i!!:!::i:;;;;;|;;;p;l;:!!;;!;;;;;;;; 

"  ^  a^±:::::::::::S::±Ji±:::a^:±x::g 
^::r  -    i  i:;::::::::lt::±l;:4:::^'T^::::x:::: 

:^--:T:::^_X:|::::::::::::_:::^i:^:S:, 

'  ^4^-^  ^  +         -    [    -  -1  '     j- + ^          -L 

=  :±::±;::4x::::::::::::"::^±^_Ji±  N| 

:x::  +  -. :  i i ,       : :  1 '    "  i  I     : 

Mm:I       I'M      ''4^^-ii^^ 

3  - '                                                                    1  1  1  M         ■  •  '  ■              1 

■pS||i;|;iig||i;|:s 

--4i-'-4--+4444--H--4--N-l— -L4j--4^-t^  ^*-^  4-^ 

X''i  i'i'  '        '1    i  iT'M  M    ml     IH;! 

-i-i---— ry-:}: i-j--|--^H ^~^ "^  -^—  VT^  + 

Tn~  \  ■ ''    ~Wz~A-ii ^^—  : '  i  i  ^ — M-l — ^ — 'r^ 

rff  n^  Tf- ^i+ ^  T^  ittt --^  - -^  -  4 

..±i±::._               M  ^^^  m     _,      ^.___. 

sji  1  imiil  II  Hill  II  III  III  II II  III  III  II  Nil  UN  H 

1^ II  1  1  N  1  1  1  1  Ijl  1  1  [M  1  1  1  1  M  1  II  1  1  1  in  1  1  1  1  1  1  1  I^IS 

parameter.     The  equation  may  then  be  written  in  the  Elimination  of  d  from  the  last  two  equations  yields 

reduced  determinant  form  the  L  curves 


-  100  cos  6 


100  cos 


0 


20 


Since  {R  —  r)  determines  0,  the  lines  parallel  to  the 
Y  axis  are  inscribed  with  the  corresponding  values  of 
This  equation  is  now  a  special  case  of  Equation  (52)       {R  —  r),  Fig.  80. 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


I   '  I  '   M   I  '   M   M  I   '  I   'I  M   '   I  '   M  M   I I   '  I 


r^ 

t 


xaaa  ao  HxoMaT 


g     ?l     S     ^ 
xaaa  ao  Hxowai 


9 

Lu_L 


S  S  S 

■  I  ■  I  I  I  I  I  I  I  I  I  I  I  ■  I  ■  I  .  I  .  I  .  i  ■  I  .  I  I  I  .  I  I 


ALIGNMENT  DIAGRAMS  WITH  ADJUSTMENT 


97 


The  L  curves  are  curves  of  translation  parallel  to 
the  Y  axis  and  were  drawn  with  a  celluloid  template. 
The  diagram  is  entered  with  the  two  values  of  the 
pulley  radii  as  decimals  of  the  shaft  center  distance 
and  the  length  L  of  the  belt  will  of  course  appear  as  a 
multiple  of  the  given  shaft  center  distance  and  will 
usually  have  to  be  changed  to  inches. 

Although  no  adjustment  of  the  index  is  necessary 
to  determine  the  values  of  L  from  the  diagram  of 
Fig.  80,  a  related  problem  for  which  the  same  diagram 
will  also  serve  does  require  adjustment  of  the  index. 
This  is  the  problem  of  finding  additional  pairs  of 
values  of  the  radii  R  and  r  (frequently  with  a  given 
ratio  K)  with  the  same  length  of  belt  and  shaft 
distance.  To  solve  this  problem  it  may  be  assumed 
that  r  =  KR,  and  the  figure  is  then  entered  with  a 
tentative  value  of  R  and  the  index  turned  to  the  inter- 
section of  the  given  L  curve  and  the  vertical  line 
marked  with  the  value  (1  —  A')  Rov  {R  —  r).  Then 
if  the  value  of  r  read  on  the  scale  for  r  is  found  to  be 
less  than  it  should  be  {i.e.  KR)  it  is  necessary  to 
select  another  value  of  R  and  repeat  the  trial  and 
several  may  be  necessary. 

It  should  be  noticed  that  if  the  parameter  K 
denoting  the  ratio  of  the  two  pulley  radii  is  introduced 
the  determinant  equation  of  the  diagram  here  given 
has  the  very  special  form  (Z  constant) 

I     1  vR  1 

-1  irKR  1 

I  m    giLR)    1 

and  the  adjustment  consists  essentially  in  finding  a 
value  of  K  for  which  the  same  value  of  R  results  in  a 
collinear  position  of  the  index. 

Example  53. — The  mean  temperature  difference 
formula  of  Professor  Greene' 

L  Ti"  -  r2"J 

may  be  written  in  the  first  determinant  form 
7(1 -» 


1 


1       T^       n 

This  formula  is  used  in  designing  heat  transfer  appa- 
ratus; the  coefficient  n  is  dependent  on  the  boundry 
conditions  and  varies  from  not  less  than  0.08  to  0.50. 
T  is  the  mean  temperature  during  the  elapsed  time 
in  which  the  temperature  difference  changes  from  Ti 
to  a  difference  of  Ti.  It  is  seen  at  once  that  a  reduced 
determinant  form  of  the  equation  obtained  by  dividing 
the  rows  by  the  elements  of  the  third  column  would 
» A.  M.  Greene,  Jr.,  "Heat  Engineering,"  McGraw-Hill  Book 
Company. 


involve  the  plotting  of  the  reciprocals  of  the  powers 
of  the  temperature  differences  Ti  and  T2  which  is  to  be 
avoided.  If  columns  one  and  two  were  combined 
for  a  new  third  column,  then  division  by  its  new 
elements  would  involve  plotting 
Ti 


1  +  Ti 


f  =  1,2 


which  would  not  give  a  good  disposition  of  the  tem- 
perature elements  as  T  varies  from  0  to  200.  How- 
ever, by  first  multiplying  column  one  by  20  there 
results  the  reduced  determinant  form 


1 

Ti 

T," 

Ti  +  20 
T, 

Ti  +  20 
T," 

T,  +  20 

T,  +  20 

The  exponent  n  enters  each  row  and  as  T  is  the 
unknown  there  will  be  no  adjustment  of  the  index. 
The  defining  equations  are 


T'd- 


ri-f20 

T2 
T2  +  20 


r, -1-20 
Ti  +  20 


There  is  a  binary  scale  on  the  line  x  =  I  but  before 
segregating  the  variables  there  involved,  an  inspection 
of  the  two  last  defining  equations  shows  that  the  lines 


*  -  r  -i-  20 
may  be  used  to  advantage;  consequently  segregate  the 
variables  n  and  T  of  the  binary  scale  by  the  equations 


X  = 


y  = 


«(i  -  xy- 


T  +  20         ^  ~     (20a;)i-" 
and  for  the  second  and  third  curve  nets  there  follows 
r,  xil  -  a;)'-" 


r, -1-20 
T2 


{20xy- 

x{i  -  xy 


"       7^2  +  20  ^  (20a;)'-" 

The  lines  parallel  to  the  Y  axis  are  to  serve  then  for 
the  values  of  the  three  temperatures  T,  Ti,  T2.  The 
diagram  shown  in  Fig.  81  was  plotted  with  the  hori- 
zontal scale  unit  equal  to  20  inches  and  the  vertical 
scale  unit  equal  100  inches.  It  is  very  suitable  for 
temperature  differences  up  to  nearly  100  degrees  and 
will  read  accurately  to  one  degree.  For  differences 
above  100  degrees  the  readings  are  less  accurate. 

There  are  necessarily  two  sets  of  w-curves.  The 
desired  value  of  T  is  found  on  the  w-curve  of  the  set 
required  in  the  binary  scale  and  at  the  intersection  of 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


ALIGNMENT  DIAGRAMS  WITH  ADJUSTMENT 


99 


a  horizontal  line  drawn  from  the  point  of  intersection 
of  the  index  with  that  scale.  The  index  is  of  course 
drawn  to  join  the  points  corresponding  to  the  given 
values  Ti,  n,  and  T^,  n. 

To  design  a  diagram  more  suited  for  temperature 
differences  between  100  and  200  degrees  it  would  be 
desirable  to  have  the  space 


200 
200  + 


100 
100  +  a 


on  the  X  axis  made  a  maximum  by  a  suitable  choice 
of  a  (which  was  chosen  equal  to  20  in  the  drawing 
here  shown).  By  equating  to  zero  the  first  derivative 
with  respect  to  a  there  results 

a  =  v^O^  =  141.4 

or  more  generally;  if  it  is  desired  to  use  the  diagram 
primarily  for  an  interval  between  Ti  =  ni  and  Jj  = 
n  it  is  advantageous  to  have  the  interval 


m  -\-  a        n  -\- 


a  maximum.    Equating  therefore  -r  to  zero,  it  is  found 

that  

a  =  yjmn 

It  is  to  be  observed  in  the  above  analysis  that  there 
will  always  be  two  sets  of  w-curves  in  the  drawing  and 
that  the  set  which  determines  T  tends  to  run  off  the 
paper  with  increasing  n  and  to  lie  near  the  X  axis  for 
small  values  of  n  and  for  large  values  of  a. 

Equations  in  five  variables  of  the  form 


/l2 

gl2 

1 

/23 

g23 

1 

/4S 

g46 

1 

=  0 


(53) 


will  now  present  no  new  difficulties. 

Example  54. — Consider  the  biquadratic  equation, 

z*  +  aiz'  +  aiz^  +  a32  +  a4  =  0 

A  first  determinant  form  of  the  equation  is, 

1  0     I 

ai  1=0 


—  a-i 
a,  +  z' 


-ai 


By  interchanging  rows  and  columns  and  dividing  the 
second  column  by  a^  there  results,  after  the  usual 
modifications,  the  reduced  determinant  equation 

—z        ai-\-  z*         1 

32  ai  , 

—  —as  1 

ai  a-i 

0  -a22'  1 


This  reduced  equation  is  a  special  case  of  Equation 
(53).     The  defining  equations  may  be  written: 
x=  -z  y  =  K(a4  +  zO 


X  =  Q  y  =  -y^a^z^ 

The  scale  factor  J^  is  needed  to  restrict  the  lengths  of 
ordinates  in  the  diagram.     The  ratio  of  the  coefficients 

—  is  regarded  as  a  parameter  K.     The  third  pair  of 

equations  determines  a  binary  scale  on  the  Y  axis.  It 
is  convenient  to  segregate  the  variables  a-i  and  z  of 
the  binary  scale  by  writing  the  equations, 

ai    , 

X  =  —z  y  =  —  tx 

The  number  of  curve  families  is  thus  reduced  since 
the  z  fines  parallel  to  the  F  axis  are  made  common  to 
the  first  and  third  curve  nets. 

The  equations  of  the  first  two  curve  nets  are 

X  =  -z  y  =  yiai  +  X*) 

X  =       K  y  —  -.X 

By  adopting  a  modulus  of  5  inches  there  can  be 
shown  on  a  diagram  20  inches  square,  the  numeri- 
cal value  of  the  roots  up  to  2.  The  a*  curves  are 
quartic  curves  of  translation  parallel  to  the  Y  axis  and 
in  the  figure  given  (Fig.  82)  were  originally  drawn 
with  a  celluloid  template.  The  parabolas  of  the  binary 
scale  determine  ordinary  scales  on  each  ordinate. 
The  diagram  is  entered  with  values  of  the  parameter 
ai 


K  =  —  and  values  of  as  which  determine  a  point  in  the 

second  curve  net.  The  index  must  then  be  rotated 
about  this  point  until  it  intersects  the  curve  marked 
with  the  given  value  of  at  on  the  same  ordinate  that 
passes  through  the  point  on  the  parabola  a^  cut  by  the 
horizontal  projecting  line  from  the  point  of  intersection 
of  the  index  and  the  Y  axis.  In  Fig.  82  the  index 
is  set  for  the  two  real  roots  of  the  biquadratic  equation 
z*  +  1.332'  +  1.62^  +  22  -  3.3  =  0 
The  equation 

fi>         g^i  1     I 

fki         gk,  1     I  =  0  (51) 

An  g^n  1 

where  the  subscripts  are  allowed  to  take  on  any 
two  different  values  in  pairs  from  the  numbers  0,  1,  2, 
3,  4,  and  5,  exhausts  all  possible  cases  of  the  equation 
in  five  variables. 


Problem  1.- 


Discuss  the  equation 

0  2^  1 

1  z  0 
a  I         -02        1 


100 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


T7m 


wm^^^mw^^^^¥mi 


I 


'Kli 


^^  ^   >'At 


m 


^^ 


^ 
^ 

^ 


ALIGNMENT  DIAGRAMS  WITH  ADJUSTMENT 


101 


0 


Problem  2. — Criticise  the  diagram  for  the  quadratic 
equation  which  results  from  the  determinant  form 

2+Ci2  -1  1 

a.,  1  1 

0  0  1 

Problem  3. — Discuss  the  case  of  equation  (51)  where 
Zi  is  unknown  and  when  the  curves  for  Zi  are  the  same  in 
two  curve  nets. 

Problem  4. — Discuss  the  possible  diagram  for  the  cubic 
equation  written  in  the  form 

0  z^  1 

1  -a^z  z 
1         fl3          -ai 

Problem  5. — The  law  of  cosines  in  trigonometry  may  be 
written  in  the  form 

—  a- 


0 


b^+c^ 
:(1  -  2b) 
-cos  A 


Construct  a  diagram. 

Problem  6. — Construct  a  diagram  for  a  formula  of 
trigonometry  which  falls  under  the  special  form  (Problem 
3)  of  Equation  (51). 

Problem  7. — The  formula 

r,  -  r. 


M 


log. 


of  Fig.  56  is  written 


T,-T, 


log. 


for   use   in   connection   with   exhaust   steam   feed   water 
heaters,  where 

Ti  =  temperature  of  the  exhaust  steam 
T2  =  temperature  of  the  water  leaving  the  heater 
To  =  temperature  of  the  water  entering  the  heater 
At  =  average  temperature  difference. 


Show  that  it  can  be  written  in  the  reduced  determinant 
form 

T,)  1 


At 
log.  (T, 


(Ti  -  Ti) 

log.  {T,  -  To) 

{Ti  -  To) 


{Ti  -  T,) 


(T,  -  To) 

and  design  a  diagram  with  one  curved  binary  scale  which 
has  two  systems  of  segregating  curves  that  serve  for  the 
three  variables  Ti,  T2,  To. 

Problem  8. — If  the  formula  of  the  preceding  problem 
is  written 

0  At 

1  (r,  -  T2) 


loge  (7-: 

1 


Ts)    log.  (Ti-Ti) 
(r,  -  To) 


log.  (Ti  -  To)    log.  (T,  -  To) 
design  the  corresponding  diagram. 

Problem  9. — The  annual  sinking  fund  which  will  accrue 
to  1  at  the  end  of  n  years  is  given  by  the  formula 

1  =  ' 

5^-(l  +  0"-l 

This  equation  may  be  given  the  determinant  form 
1 


s-„\ 

i 

0 

1 

i 
1 
1 

(1  +  i)" 

0 

a  +  0" 
1 

Identify  this  with  the  last  special  form  discussed  in  this 
chapter  and  construct  a  diagram  with  suitable  scales  for 
practical  use  for  values  of  n  between  from  5  to  20 intervals. 
Problem  10. — The  accumulation  of  an  annuity  of  1  per 
annum  at  the  end  of  n  years  is  given  as  the  formula  5-;|  = 

; This   equation   has   a   determinant  form 

similar   to   the  one  of  Problem   9.     Construct  a  useful 
diagram  for  va  ues  of  i  from  3  to  12  per  cent. 


APPENDIX  A 
DETERMINANTS  OF  THE  THIRD  ORDER 


Definition. — The  square  array  of  nine  numbers  with 
two  vertical  bars 


rtl 

bi 

Cl 

aa 

b2 

C2 

03 

63 

Ci 

is  a  convenient  symbol  for  the  expression, 

a\bnCi  +  bic^az  +  C\a-ibz  —  C1M3  —  aiCobz  —  biaiCz   (1) 

and  is  called  a  determinant  of  the  third  order.  The 
separate  letters  are  called  elements.  The  elements 
in  a  vertical  line  form  a  column  and  those  in  a  hori- 
zontal line  a  row.  The  expression  (1)  is  called  the 
expansion  of  the  determinant.  The  elements  aibic^ 
form  the  principal  diagonal  of  the  determinant  and 
the  elements  Cib-ia^  the  secondary  diagonal. 

Expansion  or  Development  of  Determinants. — 
When  the  determinant  A  is  given,  the  expansion  (1) 
may  be  obtained  as  follows:  Rewrite  the  first  and 
second  columns  to  the  right  of  the  determinant. 


The  diagonals  running  down  from  left  to  right  give 
the  positive  terms.  The  diagonals  running  down  from 
right  to  left  give  the  negative  terms.  Whenever 
negative  elements  are  present  care  must  be  taken  in 
determining  the  sign  of  each  term  in  the  expansion. 

SIMPLE  PROPERTIES  OF  DETERMINANTS 

I.  When  all  the  elements  of  one  row  or  of  one  column 
are  zero  the  value  of  the  determinant  is  zero.     This  is 


proved  by  observing  that  each  term  in  the  expansion 
contains  as  factors  one  and  only  one  element  from 
each  row  and  each  column. 

II.  //  all  the  terms  in  a  row  or  in  a  column  are  multi- 
plied {or  divided)  by  the  same  number  K,  the  value  of  the 
determinant  is  multiplied  {or  divided)  by  K.  The 
reasoning  is  the  same  as  for  I.  In  particular  if 
A"  =  —  1  the  sign  of  the  determinant  is  changed. 

III.  //  the  rows  of  a  determinant  are  changed  into 
corresponding  columns  the  determinant  is  unchanged. 
Thus 


IV.  //  two  rows  or  columns  of  a  determinant  are 
interchanged  the  sign  of  the  determinant  is  changed. 
This  property  may  be  proved  for  adjacent  rows  by 
determining  the  change  in  the  expansion  due  to  inter- 
change of  corresponding  subscripts.  Repetition  of 
this  process  will  extend  the  result  to  any  two  rows. 
By  virtue  of  III  the  result  is  true  for  columns. 

V.  If  a  determinant  has  two  rows  or  columns  identical, 
its  value  is  zero.  If  we  interchange  two  rows  we 
obtain  by  IV  —A,  but  since  the  interchange  of  identical 
rows  does  not  alter  the  determinant  we  have 


ai 

61 

Cl 

ai 

ao 

as 

ai 

62 

Cl 

= 

bi 

b. 

63 

az 

^-3 

C3 

Cl 

C2 

cz 

that  is 


A  = 
2a  = 


VI.  //  one  row  or  column  of  a  determinant  A  has  as 
elements  the  sums  of  two  or  more  numbers,  A  can  be 
written  as  the  sum  of  two  or  more  determinants. 
Thus 


1  ai  +  a/  +  a/ 

bi 

Cl 

Ol 

bi 

Cl 

Oi 

bi 

02  +  a-2   +  ai' 

b2 

C2 

a. 

62 

C2 

+ 

a/ 

b2 

\    03  +   03     +   Os" 

b3 

C3 

03 

^-3 

C3 

10. 

5 

03 

b3 

104 


DESIGN  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


VII.  The  value  of  any  determinant  A  ^5  not  changed 
if  each  element  of  any  row  or  column  multiplied  by  any 
given  number  K  be  added  to  the  corresponding  element 
of  any  other  row  or  column. 

By  II  and  VI 


b,  +  A'63        ^'2 
c\  +  Kci       Co 


Special  Properties. 

from  II  that 


02  fls 

62  ^-3        + 

Ci  Ci 


Oi 

as 

62 

b,     = 

Ci 

C3 

0-2  03 

bo         is 

Co  C3 


+  0 


-It  results,  by  V  immediately 

b,        fi     I 

bo  Co  =0 

bz        C3 


=  0 


provided  that  Ci,  C2,  C3  are  all  different  from  zero.  A 
column  of  unit  elements  may  then  always  be  intro- 
duced into  the  equation  A  =  0.  For  even  should  a 
zero  appear  in  every  column,  by  using  VII  a  column  of 
elements  all  different  from  zero  may  be  obtained  and 
by  using  IV,  this  column  may  be  given  the  third 
position.  Finally  the  determinant  may  be  divided 
by  the  elements  of  the  third  column. 

In  the  construction  of  engineering  diagrams  one  of 
the  fundamental  operations  is  to  write  certain  given 
formulas  of  three  variables  in  the  determinant 
equation  form. 


C3 


MxUtiplication  of  Determinants  of  the  Third  Order. 

The  product  of  the  two  determinants  of  the  third 
order  A  and  Ai,  is  a  determinant  of  the  third  order  as 
follows : 


OiWi  +  bini  +  Cil 
a^mi  +  btHi  +  f2l 
flsWi  +  63M1  +  C3I 


61 

fl 

nil 

Wi 

1 

62 

Co 

W2 

«2 

2 

63 

Ci 

mz 

«3 

3 

O1W2  +  bini  +  Ci2 
atnio  +  bojto  -\-  co2 
a^mo  -\-  bsno  -f  C32 

aiW3  +  bins  +  Ci3 

02^3  +   boHs  +   CoS 

a^mi  +  63M3  +  CsS 

To  prove  this  result  it  will  be  sufficient  to  actually 
carry  out  the  expansions  and  multiplications.  A 
further  proof  is  given  by  L.  G.  Weld  "Theory  of 
Determinants,"  Chapter  VI  and  in  any  work  on 
determinants. 

A  working  rule  for  multiplication  may  then  be 
stated  thus :  Connect  by  plus  signs  the  elements  of  each 
row  in  both  determinants.  Place  the  first  row  of  the 
first  determinant  upon  each  row  of  the  second  in  turn 
allowing  each  two  elements  as  they  touch  to  become 
products.  This  is  the  first  row  of  the  product.  Perform 
the  same  operation  on  the  second  determinant  with  the 
second  row  of  the  first  to  form  the  second  row  of  the 
product,  and  again  with  the  third  row  of  the  first 
determinant  to  obtain  the  third  row  of  the  product. 

Note  that  the  product  (by  virtue  of  III)  may  also 
be  obtained  by  using  columns  instead  of  rows. 


APPENDIX  B 


THE  PROJECTIVE  TRANSFORMATION 


Definition. — A  geometric  transformation  in  the 
plane  is  an  operation  which  replaces  one  geometric 
configuration  by  another.  A  one  to  one  point  trans- 
formation replaces  a  given  point  by  another  uniquely 
determined  point.  Under  the  operation  of  such  a 
transformation  the  locus  of  a  given  variable  point 
P{xy)  is  transformed  into,  or  replaced  by,  another 
definite    locus    traced    by    the    corresponding    point 

Equations  of  a  Transfonnation. — Usually  a  relation 
may  be  written  between  the  coordinates  of  a  given 
point  {xy)  and  those  of  the  transformed  point  (xiyi). 
Such  equations  are  called  the  equations  of  transforma- 
tion. Thus  for  example,  if  every  point  P  of  the  plane 
is  pushed  outward  by  an  impulse  from  the  origin  0  so 
that  the  distance  OP  is  doubled,  there  results  obviously 

^1  =  2x 

y,  =  2y 

for  the  relations  connecting  the  coordinates  of  the  old 
and  the  new  points.     Such  a  transformation  is  called 


a  dilatation.  By  it,  circles  about  the  origin  are 
transformed  into  circles  with  radii  twice  as  great. 
Straight  lines  remain  straight,  etc.  A  more  general 
dilatation  is  given  by  the  equations 

xi  =  nx  yi  =  ixy 

where  ;u  is  any  constant  whatever. 

Kinds  of  Point  Transformations.— Obviously  if  a 
pair  of  equations 

xi  =  <t>ixy)  yx  =  rPixy)  (1) 

are  written  at  will,  they  may  in  general  be  regarded  as 
establishing  geometrically  a  relation  between  the 
points  (xy)  and  the  (transformed)  points  (xiVi)  which 
may  be  computed  whenever  values  are  assigned  to 


X  and  y;  i.e.,  whenever  any  point  P  is  selected.  Now 
the  properties  of  the  resulting  geometric  transforma- 
tion will  depend  upon  the  nature  of  the  functions 
</)  and  fp  in  Equation  (1).     For  example  if 

xi  =  X  +  h 

yi  =  y 

are  the  equations,  then  every  point  of  the  plane  is 
moved  a  distance  h  parallel  to  the  X  axis  in  the  posi- 
tive direction.     A  straight  line  whose  equation  was 

Ax  +  By  +  C  =  0 
becomes 

A{x,-  k)  +  Byx  +  C  =  0 
or 

Ax,  +  By,  +  (C  -  Ah)  =  0 
which  is  obviously  another  straight  line  parallel  to 
the  first  one.  The  last  equation  is  called  the  trans- 
formed equation  and  determines  the  transformed 
locus.  To  set  up  the  equation  of  the  transformed 
locus  it  is  first  necessary  to  solve  the  equations  of  the 
transformation  for  the  variable  coordinates  x  and  y 
in  terms  of  the  coordinates  x,  and  y,  of  the  transformed 
points  and  it  will  be  assumed  here  that  this  may  always 
be  done. 

More  generally,  if  the  equations  of  a  transformation 
are 

Xi  =  aix  +  biy  +  Ci 
yi  =  a2X  +  biy  +  C2  (2) 

then  a  straight  line 

Ax  +  By  +  C  =  0  (3) 

goes  into  another  straight  line 

A,x,  +  B,y,  +  C,  =  0  (4) 

For,  solving  Equations  (2)  for  x  and  y  there  results 


-biyi  +  biXi  —  b.,ci  +  bid 


-Cl  + 

Xl 

b, 

-C2  + 

yi 

b2 

fll 

b. 

Oi 

b. 

fli 

-Ci  -1-  Xl 

02 

-C2  +   y. 

ai 

bi\ 

di 

62 

aibi  —  Oibi 


-aiXi  -\-  JiVi  —  aiC2 
+  a^ci 


Uibo  —  a^bi 


106 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


Substitution  of  these  values  of  x  and  y  in  Equation  (3) 
yields  after  collecting  coefficients,  a  linear  Equation  (4) 
in  the  new  variables,  Xiyi  in  which  Equation  Ai,Bi,Ci, 
are  expressions  involving  a,  b,  c,  only.  The  reader 
should  now  actually  make  the  necessary  substitutions 
and  prove  that  the  transformed  line  under  this 
transformation  is  always  parallel  to  the  original  line. 
The  Projective  Transfonnation. — AU  the  trans- 
formations whose  equations  have  been  written  above, 
have  the  property  that  they  transform  straight  lines 
into  straight  lines  again.  They  have  all  been  special 
cases  of  the  general  transformation 

Oix  +  biv  +  ci  OiX  +  biy  +  ci      .j., 

ozx  +  bzy  +  Cs        ^        03X  +  bzy  +  C3 
which  is  called  the  general  projective  transformolion. 
The  characteristic  of  the  equations  of  this  transforma- 
tion is  that  the  functions  4>{xy)  and  4/  {xy)  of  Equations 
(1)    are   Hnear    fractional    functions    with    the   same 


denominators. 
the  relation 


The  constant  coefficients  must  satisfy 


oi        bi        Ci    I 

a,        b,        c.     I  ^  Qi  (6) 

03  ^3  C3 
otherwise  the  coefficients  are  not  restricted.  The 
determinant  of  the  inequality  (6)  is  called  the  determi- 
nant of  the  transformation.  It  is  to  be  observed  that 
the  transformation  Equations  (2)  above  result  from 
Equations  (5)  if  03  and  bs  are  chosen  equal  to  zero  and 
C3  equal  to  unity 

Properties  of  the  Projective  Transformation.— 
There  are  two  principal  properties  enjoyed  by  this 
transformation  which  are  important  for  the  work 
needed  in  this  book.  First,  the  transformation  pre- 
serves straight  Hnes.  Solving  Equations  (5)  for  x 
and  y  it  is  found  that 

_  ^1^1  +  Biyi  +  Ci         ^  A2X1  +  Bjyi  +  Ci 
^  ~  Azx^  +  B3yi  +  C3     ^      A3X1  +  B3yi  +  C3 
where  A,  B,  C,  are  expressions  involving  a,  b,  c,  only 
and  consequently  a  straight  line 

ax+by  +  c  =  0 
becomes 
a{AiX,  +  B,y,  +  d)  +  b{A.x,  +  B.yx  +  C2)  + 

c{A3X,  +  B3y  +  C3)  =  0 
and  collecting  terms  this  equation  has  the  final  form 

o'x  +  b'y  +  c  =  ^ 
where c'  =  aAx  +  bAi.  +  CA3,  etc.,  and  is  consequently 
the  equation  of  a  new  or  transformed  straight  line. 

Second,    the    transformation    may    always    be    so 

selected  that  any  four  points  (no  three  of  which  are 

coUinear)  may  be  made  to  take  any  four  (similarly 

restricted)  positions.     This  result  is  accomplished  by 

'  Read  9^  "is  different  from." 


selecting  the  suitable  coefficients  for  the  equations  of 
the  transformation  (5).  To  prove  this  property  of 
the  projective  transformation  whose  equations  are 
written  in  the  form  (5)  above,  assume  that  the  four 
points  given  are  Pi,  Pi,  P3,  Pi,  with  the  coordinates 
(wi«i),  (nhni),  (OT3W3),  (w4«4),  or  more  briefly,  P,-  with 
coordinates  WiH,,  where  i  =  1  .  .  .4. 
Let  it  be  assumed  now  that  these  four  points  are  to  be 
tiansformed  into  the  four  new  positions  whose  coor- 
dinates are  respectively  /),g,  (j  =  1  .  .  .  4) .  There 
will  result  immediately  from  Equations  (5)  eight 
necessary  relations  of  the  form 

_  fliWi  +  biUi  +  ci  _  ajnii  +  b^nj  +  Ci   .  . 

^'  ~  flaw.  +  bini  +  C3  ^'  "  OsWi  -f  is".-  +  Cs 
which  the  nine  coefficients  a,  6,  c  of  Equations  (5)  must 
satisfy.  All  these  equations  will  be  homogeneous  in 
the  quantities  a,  b,  c  which  are  to  be  found.  There  is 
required  one  more  relation  or  equation  to  completely 
determine  the  nine  constants  and  that  relation  may 
be  selected  at  will  and  of  course  will  be  so  chosen  as 
to  reduce  the  labor  of  solving  the  equations  as  much  as 
possible. 

Example. — Suppose  that  the  four  given  points  are 
those  with  the  coordinates  (0  0),  (0  -1),  (-1  0), 
(—1  1)  and  that  it  is  desired  to  develop  a  projective 
transformation  which  will  transform  those  four 
pointsinto  the  four  points  (0  0),  (0  1),  (1  0),  (1  1), 
respectively.  Choosing  for  convenience  C3  =  1  the 
eight  equations  resulting  from  Equations  (7)  upon 
substitution  of  these  coordinate  sets  are : 
0=0 


0  = 


1  = 


1  = 


0 

C2=0 

-b,  +  c, 
-b3+  1 

-b..+ci 

^-    -b3   +    l 

-ai  +  ci 
-03+  1 

^-    -03+1 

—  oi  —  bi 

-03  —   b3 

+ 
+ 

Ci 

1 

-02   -    bi    +   C2 

^-  -a3-b3+   \ 

This  set  of  equations  reduces  at  once  to  the  set  of  four 
linear  equations 

—  fll   +    03    =     I 
-b,   +    63    =    1 

-Oi  +  03  +  bz  =  1 

-b2  +  a3  +  b3=  1 
The  solutions  are  Oi  =  1,  03  =  0,  b-i  =  —  1,  63  =  0. 
Consequently  the  equations  of  the  transformation  (5) 
become 

x  =  -X  y  =  -y 

The  important  application  of  the  above  principle 
in  the  present  volume  arises  in  connection  with  the 
selection  of  scale  factors  in  the  design  of  the  necessary 
diagrams.  Suppose  in  connection  with  a  nomogram  for 
an  equation  of  three  variables  Zi,  Z2,  Z3  it  is  desirable  to 


APPENDIX  B 


107 


move  the  2i  and  the  Zn  scales  from  the  two  parallel 
straight  lines  x  =  —I  and  x  =  \  to  the  two  lines 
X  =  —  5i  and  x  =  &2  respectively  and  at  the  same  time 
to  introduce  the  scale  factors  mi  and  ix-i  so  that  the  two 
parallel  scales  will  then  have  the  defining  equations 
X  =  -by.  y  =  Migi 

X   =^    bn  y    =    //2g2 

respectively.  In  order  to  determine  once  for  all  what 
will  be  the  nature  of  the  change  in  the  defining  equa- 
tions for  the  third  scale  it  is  only  necessary  to  observe 
that  the  change  determined  by  the  choice  of  the  two 
transformed  scales  above  is  sufficient  to  determine  a 
projective  transformation.  The  four  points  (1  0), 
(1  1),  (—1  0),  (—1  1),  have  been  transformed  respec- 
tively into  the  four  points  {bi^) Xhni) ,{—  5iO),(—  ^-ly-i)- 
Following  the  procedure  above  there  result  the  eight 
equations 


ai  +  ci 
'       az  +  f3 

a.  +  c. 
az  +  cz 

ai-\-hi-\-Ci 
'       az  +  hs  +  c 

02   +   &2   +  C2 
"'          03  +  63  +  Cz 

-ai  +  ci 

-az  +  Cz 

-ai  +  bi  +  ci 
''  ~  -az  +  bz  +  cz 

-at  -\-  bi  +  Ci 
"'  '  -az  +  bz  +  cz 

Selecting  for  convenience  the 

arbitrarily  chosen  relation 

az  +  bz  + 

cz=  1 

the  solution  of  the  nine  equations  yields 

fiibi  +  M25i           j^ 

^1^2    —    M25l 

U         Cl  =          „ 
2ai2 

02  =  0                            b,= 

M2             C2    =    0 

Ml  ~  M2                      , 

az-      2^^                   bz- 

(.                              Ml   +  M2 

and  by  substituting  these  values  in  the  Equations  (5) 
above  there  results  for  the  necessary  projective 
transformation 

(mi52  +  ti2bi)x  +  (/ii62  —  ^l■ibl) 


y  = 


-  fii)X  +  (mi  +  M2) 
2MlM2y 


(mi  —  M2)a;  +  im  +  M2) 
There  is  a  convenient  modification  of  these  equations  if 

(nibn  —  Ilibi)   =   0 

and   also  another  convenient  simplification  if   5i  = 
62  =  5  and  (/xi^i  —  m-jSo)  5^  0  from  which  results 
_     (mi  +  Ma)^  +  (mi  —  M2) 
(mi  —  M2):«;  +  (mi  +  M2) 

_     2miM2>' 

(mi  -  M2):*;  +  (mi  +  M2) 


These  are  the  equations  of  Chapter  III  numbered 
(26)-. 

The  equations  developed  for  the  introduction  of 
scale  factors  into  the  equations  numbered  (10)  and 
(13)  in  Chapter  III  may  be  obtained  by  the  method 
here  used. 

In  the  text  of  the  present  volume  the  supplementary 
transformations  that  have  been  introduced  to  better 
the  design  of  diagrams  are  all  very  simple  and  similar 
transformations  can  usually  be  selected  by  inspection ; 
it  is  desirable  to  point  out,  however,  that  in  the  design 
of  important  nomograms  for  permanent  service  the 
use  of  the  four  point  method  here  described  may  be 
the  only  way  that  the  necessary  transformation  can 
be  determined. 

It  is  obvious  from  Equations  (5)  that  if  a  point  P 
with  the  coordinates  {m  n)  is  to  be  transformed  to 
infinity  it  is  only  necessary  to  choose  azbzCz  so  that 
aztn  +  bznix  +  Cz  =  0,  since  then  the  values  of  both 
Xi  and  yi  will  be  infinite.  The  equations  of  trans- 
formation numbered  (2)  above  are  the  most  general 
equations  for  the  projective  transformation  which 
preserves  parallelism  of  straight  lines.  Such  projective 
transformations  are  called  affine  transformations. 

The  Projective  Transformation  and  Determinants. 
The  condition  that  three  points  {x'y'),  {x"y"), 
{x"'y"')  shall  lie  on  a  straight  line  is  conveniently 
expressed  in  the  form 

:'  V  1 

■"  y"  1 

:'"  y'"  1 

If  a  general  projective  transformation  is  applied  to  all 
the  points  in  the  plane  the  three  points  in  question  go 
over  into  three  new  points  which  are  collinear  also. 
Substituting  for  x'  and  y' ,  etc.,  in  the  above  determi- 
nant the  corresponding  values  obtained  above  from 
Equations  (5)  in  terms  of  x\  and  y'l,  etc.,  there  results 


A  3X1'  +  Bzyi'  +  Cz      AzXi  +  Bzji  +  Cz 
AxXx' ■\- Bxyx' ■\- Cx    A2X1"  +  B2yx"  +  C2 


Axx'  +  Bxv'  +  Cx  Aox'  +  B.y' 
Axx"  +  Bxy"  -f- Cl  A2X"  +  B-iy" 
Axx'"  -{■  Biy'"  -f  Cl     Aix'"  -\-  B^y" 


AzXi"  +  Bzyi"  +  C3    AzXx"  +  Bzyi"  +  C3 
^1X1'"  +  Bxyx"'+  Cx  A2Xx"' +  B^yi'"  +  C2 


AzXxx"'+  Bzyx'"  +  Cz  AzXx"  +  Bzyx"  +  C3 

and  multiplying  this  equation  by  the  three  denomi- 
nators of  the  elements  of  the  first  column,  there 
results 

-f  C2     Azx'    +  Bzy'     +  C: 

-t-  Co    Azx"  +  Bzy"   +  C: 

'+C2     Az^'"  ^- Bzy"'  +  C. 


108 


DESIGN  OF  DIAGRAMS  FOR  ENGINEERING  FORMULAS 


which  by  the  multiplication  law  of  determinants  is 


1 

Bi 

Ci 

Xi' 

y\ 

1 

Bi 

c. 

X 

Xi" 

y" 

1 

3 

B^ 

C^ 

Xi'" 

y\" 

1 

Since  now  the  first  determinant  factor  does  not  vanish^ 
the  second  must  and  hence  the  condition  that  the 
transformed  points  lie  also  upon  a  straight  line 
appears  at  once  as  a  result  of  their  original  coUinear 
position. 

If  now  it  is  desired  to  write  the  above  equa- 
tion in  terms  of  the  original  coordinates  there 
follows: 


Xx 

y\ 

1 

Cl 

&1 

Cl 

X\" 

yi" 

1 

= 

^2 

^2 

Cl 

Xx" 

y'i 

1 

^3 

bz 

cz 

Which  may  be  proved  by  the  laws  of  multiplication  of 
determinants  and  Equations  (5). 

There  results  then  the  Working  Rule: 

To  apply  a  projective  transformation  to  the  variable 
elements  of  a  determinant  multiply  the  determinant  by 
the  determinant  of  the  transformation.  This  rule  may 
be  used  as  a  check  in  the  practice  involved  in  this 
voLume.  The  important  principle,  however,  which  the 
above  rule  brings  out  is  in  connection  with  the  manip- 
ulation of  first  determinant  equations  to  reduce  them : 
Every  manipulation  of  a  determinant  equation  by  the 
laws  of  determinants  is  equivalent  to  applying  to 
its  elements  a  projective  transformation.  In  other 
words  every  change  in  the  first  determinant  form  has 
corresponding  to  it  a  geometric  change  in  the  plane. 


y'        1 

y"  1        =0 

y'"      1    I 


be  shown  to  be  true  from  the  condition  VI. 


INDEX 


Accuracy,  choice  of  units  for,  72 

of  a  scale,  8 
Adjustment  diagrams,  alignment,  88 
AfBne  transformations,  107 
Air  compression,  intercooler  pressures,  64 

horsepower,  33 
Alignment  diagrams,  definition,  35 
Amortization  factor,  bonds,  75 
Anamorphosis,  13 

Angular  distance  of  celestial  body  from  meridian,  74 
Annuity  formula,  101 
Automobile  engine  rating,  43 

radiation  reduction  formula,  87 

tractive  resistance,  64 
Auxiliary  variable,  76 

variables,  hexagonal  diagrams,  30 
parallel  indices,  84 


Barometer  readings,  corrected,  39 

Bazin's  formula,  69 

Belt  tensions,  30 

Binary  scale,  curve  support  for,  92 

defining  equations  for,  66 

definition  of,  65 

methods  of  plotting  curve  nets,  74 
Binary  scales,  segregation  of  variables,  examples  of,  97,  99 

use  of  in  diagrams  with  adjustment,  90 
Biquadratic  equation,  99 
Bond  formula,  72,  75 
Boussinesq's  formula,  30 
Brauer's  method,  17 


Canals,  mean  hydraulic  radius  of  trapezoidal  sections,  55, 

64,  75,  76 
Celluloid,  use  of  transparent  with  indices,  21,  83 
Change  of  scale  factor,  2 

Chezy's  formula  for  flow  of  water  in  open  channels,  78 
Chimney  formula,  49 
Choice  of  Scale  Factor,  7 
Circles  and  straight  lines,  diagrams  of,  30 
Circular  segment,  approximate  area,  18,  64 

exact  area,  39 

mean  hydraulic  radius  of,  64 


Coefficients  as  scale  factors,  28 
Collinear  nomogram  definition,  35 

points,  36 
CoUineation  of  three  points,  35 
Colors  used  to  simplify  diagrams,  74 
Column  formula,  Gordon's,  87 

in  a  determinant,  103 
Combinations  of  simple  and  collinear  diagrams,  80 

of  simple  diagrams  for  four  variables,  80 
Common  normal,  system  of  parallel  lines,  21 
Complete  cubic  equation,  67',  75 
Compound  interest,  30,  64 
Compressed  air,  mean  effective  pressure,  30 

horsepower,  33 
Condenser  tubes,  34 
Cone  pulleys,  open  belt,  97 
Cooper's  formula,  32 
Coordinates  of  ray  center,  3 
Corresponding   values   of   the   variable,   in   diagrams   of 

adjustment,  88 
Cosines,  law  of,  trigonometry,  101 
Cubic  equation,  51 

complete  equation,  67,  75 

general  equation,  94 
Curve  nets,  65 

method  of  plotting  one  set  of  curves  by  intersections,  74 

when  one  set  of  curves  become  straight  lines,  74,  75 

with  adjustment,  88 
Curve  support  for  a  binary  scale,  92 
Curved  binary  scale,  67 

scales,  49 
Curves  of  translation,  97 

transformed  to  straight  lines,  13 


Defining  equations,  definition,  35 

for  the  binary  scale,  66 
Derivation  of  new  scales,  1 
Deriving  a  scale  factor,  method  of,  47 
Development  of  determinants,  103 
Determinant  of  the  transformation,  106 
Determinants,  effect  of  manipulating,  38 

example  of  how  to  set  up,  49 

of  the  third  order,  103 

properties  of,  103 
Determining  unknown  exponents  from  empirical  formula, 

14 
Diagonals,  of  determinants,  103 


110 


INDEX 


Diagrams,  alignment,  with  two  or  more  indices,  76 

collinear,  with  two  parallel  scales  and  one  curve  net,  66 

of  adjustment,  when  no  adjustment  is  necessary,  91 

of  ahgnment  with  one  fixed  point,  57 

with  adjustment,  88 

with  three  parallel  straight  scales,  36 
Dilatation,  projective  transformation,  105 
Displacement  of  parallel  scales,  43 
Double  alignment  diagrams,  76 

coUineation,  76 

graduation,  points  of,  65 


Hexagonal  diagrams,  28 

for  n  variables,  30 
Hinge  scale,  76,  83 
Horizontal  formula,  stadia  distance,  53 


Inductive  reactance,  34 

voltage,  28 
Intercooler  pressures,  in  air  compression,  64 


Earthwork  computations,  36,  64 
Eckblaw's  silo  formula,  30 
Elementary  diagrams,  9 
Elements,  of  a  determinant,  103 
Ellipse,  perimeter  of,  30 
End  areas  in  earthwork,  36,  64 
Expansion  of  determinants,  103 
External  of  two  tangents,  14 
Equal  ordinary  scales  on  both  axes,  11 
Equation  in  three  variables,  eight  general  forms,  94 
Equations  in  more  than  three  variables,  diagrams  with 
adjustment,  94 
in  two  variables,  5 
Equilateral  hyperbolas,  75 


First  determinant  form,  definition,  36 

rule  for  obtaining,  51 
Five  variables,  equations  in,  99 
Flow  of  water  {see  also  Francis,  reclamation  service),  25 

Bazin's  formula,  69 

Chezy's  formula,  78 

Kutter's  formula,  67 
Flynn's  modification  of  Kutter's  formula,  67 
Four  parallel  straight  scales  for  four  variables,  61,  64 
Four-point  method  in  projective  transformations,  106 

systems  of  parallel  straight  lines,  25 

variables,  diagrams  for,  with  parallel  or  perpendicular 
indices,  80 
with  two  parallel  scales  and  a  net,  66 
Francis'  weir  formula,  11,  21,  47,  57,  62 
Friction  head,  flow  of  water,  21 

factor  for  steam  turbine  nozzles,  25 
Function  scale,  1 


Kutter's  formula,  67 


Lame's  formula  for  thick  cylinders,  80 
Law  of  cosines,  trigonometry,  101 
Length  of  a  scale  between  limits,  7 
Length  of  open  belt,  stepped  pulleys,  94 
Lewis  formula  for  gear  teeth,  87 
Limiting  values  of  the  variable,  8 
Locus,  of  transformed  points,  104 
Logarithmic  cross-section  paper,  14,  57,  60 

transformation,  38 
Log  tan  z  from  log  z  scale,  5 


M 


Machining  time,  84 

Manipulating  determinants,  103 

Manipulation  of   determinants  equivalent   to  projective 

transformation,  108 
Mean  efifective  pressure  of  expanding  steam,  14 
hydraulic  radius,  circular  segment,  64 

trapezoidal  section,  55,  64,  75,  76 
temperature  difference  by  Greene's  formula,  97 
log  formula,  63,  101 
Modulus,  1 

Multiplication  diagram,  62 
of  determinants,  104 


N 


New  scales,  methods  of  deriving  from  given  scales,  1 
Nomogram,  9 
Non-parallel  scales,  49 
Normal  form,  21 


Gear  teeth,  Lewis  formula  for,  87 

Gordon's  column  formula,  87 

Grashof  formula,  30,  43 

Greene's  formula,  mean  temperature  difference,  97 

H 

Heat  drop,  adiabatic,  25 

transfer  apparatus,  mean  temperature  difference, 


Oblique  axes,  use  of,  39 

scales,  49 
One  fixed  point  in  diagrams  of  alignment,  57 
Open  belt  formula,  94 
Ordinary  scale,  1 
Orifice,  flow  of  steam  through,  43 

rectangular,  flow  of  water  under  low  head,  62 


INDEX 


111 


Parallel  indices,  80 

straight  lines,  21 
Parameter,  9 
Parametric  equations,  35 
Partial  differentiation,  36 
Perpendicular  indices,  80 
Pitch,  gear  teeth,  circular  and  diametral,  87 
Pivot  scale,  use  of,  76 
Plotting  curves  by  dividing  ordinates,  9,  13 

function  scale  from  graph  of  curve,  5 
Point  transformations,  105 
Points  of  double  graduation,  65 
Principal  diagonal  of  a  determinant,  103 
Projective  scale,  2 

transformation,  104 
effect  of,  38 
Properties  of  determinants,  103 

the  projective  transformation,  106 
Purchase  price  of  bonds,  72 
P7"  =  constant,  17,  60,  64 


Segment  of  circle,  approximate  area,  18 

exact  area,  39 
Selection  of  curves  in  net  of  binary  scale,  69,  74 
Shaft  diameter,  to  transmit  given  h.p.  and  r.p.m.,  17 

for  combined  bending  and  twisting,  31 
Simple  Cartesian  diagrams,  9 

diagrams,  combined  for  four  variables,  80 

straight  line  diagrams,  13 
Singular  point,  69 
Sinking  fund  formula,  101 
Special  forms  of  equations,  for  diagrams  of  adjustment, 

properties  of  determinants,  104 
Stadia  formulas,  53 

Steam,  Unwin's  formula  for  flow  of  in  pipes,  77 
Stepped  pulleys,  open  belt,  94 
Straight  lines  and  circles,  diagrams  of,  30 

scales,  three  parallel,  diagrams  for,  36 

two  parallel  one  oblique,  diagrams  for,  43 
Superimposed  diagram,  76 
Supplementary  transformations,  106 
Support,  of  binary  function  scale,  65 


Quadratic  equation,  12,  49 
diagram  with  adjustment 
Quartic  curve,  55 


90 


Radial  lines,  how  to  plot,  16 

line  systems,  14 
Ratio  of  expansion,  16 
Ray  center,  3 

Reciprocal  scales,  methods  of  avoiding  use  of,  50,  97 
Reclamation  service  formula  for  flow  of  water,  21,  28,  61, 

62 
Rectangular  orifice,  flow  of  water,  low  head,  62, 
Redemption  price  of  bonds,  72,  75 
Reduced  determinant  form,  36 
Richardson's  equation,  34 
Riveted  joints,  10 
Row,  in  a  determinant,  103 
Rules  concerning  determinants,  103 


Taylor's  formula  for  tool  pressure,  30,  39 
Thermionic  current,  34 
Thermodynamics  equations,  64 
Thick  cylinders,  Lame's  formula  for,  80 
Third  order  determinants,  103 

scale,  scale  factor,  38 
Three  curve  nets,  collinear  diagrams  for,  74 

parallel  straight  scales,  36 
at  unequal  distances,  38 

straight  scales,  no  two  parallel,  49 

variables,  most  general  form  of  equation,  93 
Tool  pressure,  Taylor's  formula,  30,  39 
Torus,  volume  of,  39 
Transformation,  projective,  105 
Transformations,  affine,  107 
Trapezoidal  canals,  m.h.r.,  55,  64,  75,  76 
Turbine  nozzle,  25 
Two  variable  equations,  5 


U 


Unwin's  formula,  flow  of  steam  in  pipes,  77 


Scale  factor,  definition,  1 
choice  of,  7 

development   of,   for   three   parallel  scales  at  unequal 
distances,  38 
for  two  straight  scales  at  right  angles  and  one  curve 
scale,  55 
for  Cartesian  diagrams,  9 
Scale  of  log  z,  1 

of  Vz,  1 
Scales  determined  by  curves  on  parallel  straight  lines,  67 
Secondary  diagonal  of  a  determinant,  103 


Vertical  distance,  stadia,  formula  53, 
Volume  of  frustum  of  cone,  30 


W 


Water,  flow  of,  Kutter's  formula  {see  also  Francis,  reclama- 
tion service),  67 
Bazin's  formula,  69 
Wind  resistance  of  automobiles,  28 


